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  <doc-data|<doc-title| The <math|\<Phi\><rsup|4><rsub|3>> measure via
  Girsanov's theorem>|<doc-author|<author-data|<author-name|N.<nbsp>Barashkov
  and M.Gubinelli>|<\author-affiliation>
    Hausdorff Center of Mathematics &

    Institute of Applied Mathematics

    University of Bonn, Germany
  </author-affiliation>>>>

  <abstract-data|<abstract|We construct the <math|\<Phi\><rsup|4><rsub|3>>
  measure on a periodic three dimensional box as an absolutely continuous
  perturbation of a random shift of the Gaussian free field. The shifted
  measure is constructed via Girsanov's theorem and the relevant filtration
  is the one generated by a scale parameter. As a byproduct we give a
  self-contained proof that the <math|\<Phi\><rsup|4><rsub|3>> measure is
  singular wrt. the Gaussian free field.>|<abstract-msc|60H30|81T08>|<abstract-keywords|Constructive
  Euclidean quantum field theory, Boué\UDupuis formula, paracontrolled
  calculus.>>

  <section|Introduction>

  The <math|\<Phi\><rsup|4><rsub|3>> measure on the three dimensional torus
  \ <math|\<Lambda\>=\<bbb-T\><rsup|3>=<around*|(|\<bbb-R\>/2\<pi\>\<bbb-Z\>|)><rsup|3>>
  is the probability measure <math|\<nu\>> on distributions
  <math|<value|CS><rprime|'><around*|(|\<Lambda\>|)>> corresponding to the
  formal functional integral

  <\equation>
    \<nu\><around*|(|\<mathd\>\<varphi\>|)>=<lprime|``><around*|{|<frac|1|Z>exp<around*|[|-\<lambda\><big|int><rsub|\<Lambda\>><around*|(|\<varphi\><rsup|4>-\<infty\>\<varphi\><rsup|2>|)>\<mathd\>x|]>\<mu\><around*|(|\<mathd\>\<varphi\>|)>|}><rprime|''><label|eq:phi43>
  </equation>

  where <math|\<mu\>> is the law of the Gaussian free field with covariance
  <math|<around*|(|1-\<Delta\>|)><rsup|-1>> on <math|\<Lambda\>>, <math|Z> a
  normalization constant and <math|\<lambda\>> a coupling constant. The
  <math|\<infty\>> appearing in this expression reminds us that many things
  are wrong with this recipe. The key difficulty can be traced to the fact
  that the measure we are looking for it is not absolutely continuous wrt.
  the reference measure <math|\<mu\>>. This fact seems part of the folklore
  even if we could not find a rigorous proof for it in the available
  literature apart from a work of Albeverio and
  Liang<nbsp><cite|albeverio_remark_2008> which however refers to the
  Euclidean fields at time zero. The singularity of the
  <math|\<Phi\><rsup|4><rsub|3>> measure is indeed a major technical
  difficulties in a rigorous study. Obtaining a complete construction of this
  formal object (both in finite and infinite volume) has been one of the main
  achievements of the constructive quantum field theory
  program<nbsp><cite|glimm_positivity_1973|feldman_lambda_1974|park_lambda_1975|feldman_wightman_1976|magnen_infinite_1976|benfatto_ultraviolet_1980|brydges_new_1983>.\ 

  \;

  In recent years the rigorous study of the <math|\<Phi\><rsup|4><rsub|3>>
  model has been pursued from the point of view of
  <with|font-shape|italic|stochastic quantization>. In the original
  formulation of Parisi\UWu<nbsp><cite|parisi_perturbation_1981>, stochastic
  quantization is a way to introduce additional degrees of freedom (in
  particular a dependence on a fictious time) in order to obtain an
  <with|font-shape|italic|equation> whose solutions describe a measure of
  interest, in this case the <math|\<Phi\><rsup|4><rsub|3>> measure on
  <math|\<Lambda\>> as in<nbsp><eqref|eq:phi43> or its counterpart in the
  full space. Rigorous analysis of stochastic quantization for simpler models
  like <math|\<Phi\><rsup|4><rsub|2>> (the two-dimensional analog of
  eq.<nbsp><eqref|eq:phi43>) \ started with the
  work<nbsp><cite|jona_lasinio_stochastic_1985>. It has been only with the
  fundamental work of Hairer on regularity
  structures<nbsp><cite|hairer_theory_2014> that the three dimensional model
  could be successfully attacked, see also<nbsp><cite|catellier_paracontrolled_2013|kupiainen_renormalization_2016>.
  This new perspective on this and related problems led to a series of new
  results on the global space-time control of the stochastic
  dynamics<nbsp><cite|MWcomedown|gubinelli_global_2019|albeverio_invariant_2017|moinat_space_time_2018>
  and to a novel proof of the constructions of non-Gaussian Euclidean quantum
  field theories in three dimensions<nbsp><cite|gubinelli_pde_2018>.\ 

  \;

  A conceptual advantage of stochastic quantization is that it is insensitive
  to questions of absolute continuity wrt. to a reference measure. This, on
  the other hand, is the main difficulty of the Gibbisan point of view as
  expressed in eq.<nbsp><eqref|eq:phi43>. In order to explore further the
  tradeoffs of different approaches we have recently developed a variational
  method<nbsp><cite|barashkov_gubinelli_variational> for the construction and
  <with|font-shape|italic|description> of <math|\<Phi\><rsup|4><rsub|3>> with
  which we were able to provide an explicit formula for the Laplace transform
  of <math|\<Phi\><rsup|4><rsub|3>> in terms of a stochastic control problem.
  In this control problem the controlled process represent the scale-by-scale
  evolution of the interacting random field.\ 

  \;

  The present paper is the occasion to explore further this point of view by
  constructing a novel measure via a random shift of the Gaussian free field
  and proving that the <math|\<Phi\><rsup|4><rsub|3>> measure can be
  constructed as an absolutely continuous perturbation thereof. Without
  entering into technical details now let us give the broad outline of this
  construction. We consider a Brownian martingale
  <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> with values in
  <math|<value|CS><rprime|'><around*|(|\<Lambda\>|)>> and such that
  <math|W<rsub|t>> is a regularization of the Gaussian free field
  <math|\<mu\>> at scale <math|t>. Let us denote <math|\<bbb-P\>> its law,
  <math|\<bbb-E\>> the corresponding expectation. In particular
  <math|W<rsub|t>\<rightarrow\>W<rsub|\<infty\>>> as
  <math|t\<rightarrow\>\<infty\>> and <math|W<rsub|\<infty\>>> has law
  <math|\<mu\>>. We can identify the <math|\<Phi\><rsup|4><rsub|3>> measure
  <math|\<nu\>> as the weak limit <math|\<nu\><rsup|T>\<rightarrow\>\<nu\>>
  as <math|T\<rightarrow\>\<infty\>> of the family of probability measures
  <math|<around*|(|\<nu\><rsup|T>|)><rsub|T\<geqslant\>0>> defined as

  <\equation*>
    \<nu\><rsup|T><around*|(|A|)>=\<bbb-P\><rsup|T><around*|(|W<rsub|T>\<in\>A|)>,
  </equation*>

  where <math|\<bbb-P\><rsup|T>> is the measure on paths
  <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> with density

  <\equation*>
    <frac|\<mathd\>\<bbb-P\><rsup|T>|\<mathd\>\<bbb-P\>>=<frac|1|Z<rsub|T>>e<rsup|-V<rsub|T><around*|(|W<rsub|T>|)>>,
  </equation*>

  and\ 

  <\equation*>
    V<rsub|T><around*|(|\<varphi\>|)>\<assign\>\<lambda\><big|int><rsub|\<Lambda\>><around*|(|\<varphi\><around*|(|x|)><rsup|4>-a<rsub|T>\<varphi\><around*|(|x|)><rsup|2>+b<rsub|T>|)>\<mathd\>x,
  </equation*>

  is a quartic polynomial in the field <math|\<varphi\>> with
  <math|<around*|(|a<rsub|T>,b<rsub|T>|)><rsub|T>> a family of (suitably
  diverging) renormalization constants. The presence of the scale parameter
  <math|t\<in\>\<bbb-R\><rsub|+>> allows to introduce a filtration and a
  family of measures <math|\<bbb-Q\><rsup|v>> defined as the Girsanov
  transformation

  <\equation>
    <around*|\<nobracket\>|<frac|\<mathd\>\<bbb-Q\><rsup|v>|\<mathd\>\<bbb-P\>>|\|><rsub|<value|CF><rsub|T>>=exp<around*|(|L<rsub|T><rsup|v>-<frac|1|2><around*|\<langle\>|L<rsup|v>|\<rangle\>><rsub|T>|)>,<space|2em>L<rsup|v><rsub|t>=<big|int><rsub|0><rsup|t><around*|\<langle\>|v<rsub|s>,\<mathd\>W<rsub|s>|\<rangle\>><rsub|L<rsup|2><around*|(|\<Lambda\>|)>><label|eq:girs>
  </equation>

  where <math|<around*|(|<around*|\<langle\>|L<rsup|v>|\<rangle\>><rsub|t>|)><rsub|t\<geqslant\>0>>
  is the quadratic variation of the (scalar) local martingale
  <math|<around*|(|L<rsup|v><rsub|t>|)><rsub|t\<geqslant\>0>> and
  <math|<around*|(|v<rsub|t>|)><rsub|t\<geqslant\>0>> is an adapted process
  with values in <math|L<rsup|2><around*|(|\<Lambda\>|)>>. Let\ 

  <\equation*>
    D<rsub|T>\<assign\><frac|1|Z<rsub|T>>e<rsup|-V<rsub|T><around*|(|W<rsub|T>|)>><around*|(|<frac|\<mathd\>\<bbb-Q\><rsup|v>|\<mathd\>\<bbb-P\>>|)><rsup|-1>,
  </equation*>

  be the density of <math|\<bbb-P\><rsup|T>> wrt. <math|\<bbb-Q\><rsup|v>>.
  We will show that it is possible to choose <math|v> in such a way that the
  family <math|<around*|(|D<rsub|T>|)><rsub|T\<geqslant\>0>> is bounded in
  all <math|L<rsup|p><around*|(|\<bbb-Q\><rsup|v>|)>> and that
  <math|D<rsub|T>\<rightarrow\>D<rsub|\<infty\>>> in <math|L<rsup|1>>. With
  particular choice of <math|v> we call <math|\<bbb-Q\><rsup|v>> the
  <with|font-shape|italic|drift> measure: it is the central object of this
  paper. By Girsanov's theorem the canonical process
  <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> satisfies the equation

  <\equation*>
    \<mathd\>W<rsub|t>=v<rsub|t>\<mathd\>t+\<mathd\><wide|W|~><rsub|t>,<space|2em>t\<geqslant\>0,
  </equation*>

  where <math|<around*|(|<wide|W|~><rsub|t>|)><rsub|t\<geqslant\>0>> is a
  Gaussian martingale under <math|\<bbb-Q\><rsup|v>> (and has law equal to
  that of <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> under
  <math|\<bbb-P\>>, that is is a regularized Gaussian free field). We will
  show also that the drift <math|v<rsub|t>> can be written as a (polynomial)
  function of <math|><math|<around*|(|<wide|W|~><rsub|s>|)><rsub|s\<in\><around*|[|0,t|]>>>,
  that is <math|v<rsub|t>=V<rsub|t><around*|(|<around*|(|<wide|W|~><rsub|s>|)><rsub|s\<in\><around*|[|0,t|]>>|)>>.
  Therefore we have an explicit description of the process
  <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> under the drift measure
  <math|\<bbb-Q\><rsup|v>> as the unique solution of the path-dependent SDE

  <\equation>
    \<mathd\>W<rsub|t>=V<rsub|t><around*|(|<around*|(|<wide|W|~><rsub|s>|)><rsub|s\<in\><around*|[|0,t|]>>|)>\<mathd\>t+\<mathd\><wide|W|~><rsub|t>,<space|2em>t\<geqslant\>0.<label|eq:sde-girs>
  </equation>

  Let us note that this formula expresses the \Pinteracting\Q random field
  <math|<around*|(|W<rsub|t>|)><rsub|t>> as a function of the \Pfree\Q field
  <math|<around*|(|<wide|W|~><rsub|t>|)><rsub|t>>. In this respect is a
  formula with very similar technical merits as the stochastic quantization
  approach.

  \;

  Intuitively this new measure <math|\<bbb-Q\><rsup|v>>, is half way between
  the variational description in<nbsp><cite|barashkov_gubinelli_variational>
  and the (formal) Gibbsian description of eq.<nbsp><eqref|eq:phi43>. It
  constitutes a measure which is relatively explicit, easy to construct and
  analyze and which can be used as reference measure for
  <math|\<Phi\><rsup|4><rsub|3>>, very much like the Gaussian free field can
  be used as reference measure for <math|\<Phi\><rsup|4><rsub|2>><nbsp><cite|glimm_quantum_1987>.\ 

  \;

  As an application we provide a self-contained proof of the singularity of
  the <math|\<Phi\><rsup|4><rsub|3>> measure <math|\<nu\>> wrt. the Gaussian
  free field <math|\<mu\>>. \ As we already remarked the singularity of
  <math|\<Phi\><rsup|4><rsub|3>> seems to belongs to the folklore and we were
  not able to trace any written proof of that. However M.<nbsp>Hairer, during
  a conference at Imperial College in 2019 showed us a unpublished proof of
  him of singularity using the stochastic quantization equation. Our proof
  and his are very similar and we do not claim any essential novelty in this
  respect. Albeit the proof is quite straightforward we wrote down all the
  details in order to provide a reference for this fact. The main point of
  the present paper remains that of describing the drift measure as a novel
  object in the context of <math|\<Phi\><rsup|4><rsub|3>> and similar
  measures.

  \;

  Our proof of singularity, in particular this also shows that the drift
  measure <math|\<bbb-Q\><rsup|v>> is singular wrt. <math|\<bbb-P\>>. The
  intuitive reason is that the drift <math|<around*|(|V<rsub|t>|)><rsub|t\<geqslant\>0>>
  in the SDE<nbsp><eqref|eq:sde-girs> is not regular enough (as
  <math|t\<rightarrow\>\<infty\>>) to be along Cameron\UMartin directions for
  the law <math|\<bbb-P\>> of the process
  <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> and therefore the
  Girsanov transform<nbsp><eqref|eq:girs> gives a singular measure when
  extended all the way to <math|T=+\<infty\>>.

  \;

  \ 

  <no-indent><with|font-series|bold|Acknowledgments.> M.G. would like to
  thank S. Albeverio, D.<nbsp>Brydges, C. Garban and M.<nbsp>Hairer for
  interesting discussions on the topic of singularity of
  <math|\<Phi\><rsup|4><rsub|3>>. The authors would like to thank the Isaac
  Newton Institute for Mathematical Sciences for support and hospitality
  during the program SRQ: Scaling limits, Rough paths, Quantum field theory
  during which part of the work on this paper was undertaken. This work is
  supported by DFG via CRC 1060 and by EPSRC via Grant Number EP/R014604/1.

  \;

  <no-indent><with|font-series|bold|Notations.> Let us fix some notations and
  objects.\ 

  <\itemize-dot>
    <item>For <math|a\<in\>\<bbb-R\><rsup|d>> we let
    <math|<around*|\<langle\>|a|\<rangle\>>\<assign\><around|(|1+<around*|\||a|\|><rsup|2>|)><rsup|1/2>>.

    <item>The constant <math|\<varepsilon\>\<gtr\>0> represents a small
    positive number which can be different from line to line.

    <item>Denote with <math|<CS><around*|(|\<Lambda\>|)>> the space of
    Schwartz functions on <math|\<Lambda\>> and with
    <math|<value|CS><rprime|'><around*|(|\<Lambda\>|)>> the dual space of
    tempered distributions. The notation <math|<wide|f|^>> or
    <math|<value|CF>f> stands for the space Fourier transform of <math|f> and
    we will write <math|g<around*|(|\<mathD\>|)>> to denote the Fourier
    multiplier operator with symbol <math|g:\<bbb-R\><rsup|n>\<rightarrow\>\<bbb-R\>>,
    i.e. <math|<value|CF><around*|(|g<around*|(|\<mathD\>|)>f|)>=g*<value|CF>f>.

    <item><math|B<rsup|\<alpha\>><rsub|p,q>=B<rsup|\<alpha\>><rsub|p,q><around*|(|\<Lambda\>|)>>
    denotes the Besov spaces of regularity <math|\<alpha\>> and integrability
    indices <math|p,q> as usual. <math|<value|VV><rsup|\<alpha\>>=<value|VV><rsup|\<alpha\>><around*|(|\<Lambda\>|)>>
    is the Hölder\UBesov space <math|B<rsup|\<alpha\>><rsub|\<infty\>,\<infty\>>>,
    <math|W<rsup|\<alpha\>,p>=W<rsup|\<alpha\>,p><around*|(|\<Lambda\>|)>>
    denote the standard fractional Sobolev spaces defined by the norm
    <math|<around*|\<\|\|\>|f|\<\|\|\>><rsub|W<rsup|s,q>>\<assign\><around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|s>f|\<\|\|\>><rsub|L<rsup|q>>>
    and <math|H<rsup|\<alpha\>>=W<rsup|\<alpha\>,2>>. The symbols
    <math|\<prec\>,\<succ\>,\<circ\>> denotes spatial paraproducts wrt. a
    standard Littlewood\UPaley decomposition. The reader is referred to
    Appendix<nbsp><reference|sec:appendix-para> for an overview of the
    functional spaces and paraproducts.
  </itemize-dot>

  <section|The setting>

  The setting of this paper is the same of that of our variational
  study<nbsp><cite|barashkov_gubinelli_variational>. In this section we will
  briefly recall it and also state some results from that paper which will be
  needed below. They concern the Boué\UDupuis formula and certain estimates
  which will be important also in our analysis of absolute continuity.\ 

  \;

  Let <math|\<Omega\>\<assign\>C<around*|(|\<bbb-R\><rsub|+>;<value|VV><rsup|-3/2-\<varepsilon\>><around*|(|\<Lambda\>|)>|)>>
  and <math|<CF>> be the Borel <math|\<sigma\>>\Ualgebra of <math|\<Omega\>>.
  On <math|<around*|(|\<Omega\>,<CF>|)>> consider the probability measure
  <math|\<bbb-P\>> which makes the canonical process
  <math|<around*|(|X<rsub|t>|)><rsub|t\<geqslant\>0>> a cylindrical Brownian
  motion on <math|L<rsup|2><around*|(|\<Lambda\>|)>> and let
  <math|<around*|(|<CF><rsub|t>|)><rsub|t\<geqslant\>0>> the associated
  filtration. In the following <math|\<bbb-E\>> without any qualifiers will
  denote expectations wrt. <math|\<bbb-P\>> and
  <math|\<bbb-E\><rsub|\<bbb-Q\>>> will denote expectations wrt. some other
  measure <math|\<bbb-Q\>>.\ 

  On the probability space <math|<around*|(|\<Omega\>,<CF>,\<bbb-P\>|)>>
  there exists a collection <math|<around|(|B<rsub|t><rsup|n>|)><rsub|n\<in\><around*|(|\<bbb-Z\>|)><rsup|d>>>
  of complex (2-dimensional) Brownian motions, such that
  <math|<wide|B<rsup|n><rsub|t>|\<bar\>>=B<rsup|-n><rsub|t>>,
  <math|B<rsup|n><rsub|t>,B<rsup|m><rsub|t>> independent for
  <math|m\<neq\>\<pm\>n> and <math|X<rsub|t>=<big|sum><rsub|n\<in\>\<bbb-Z\><rsup|d>>e<rsup|i<around*|\<langle\>|n,\<cdot\>|\<rangle\>>>B<rsup|n><rsub|t>>.

  \;

  Fix some <math|\<rho\>\<in\>C<rsub|c><rsup|\<infty\>><around*|(|\<bbb-R\><rsub|+>,\<bbb-R\><rsub|+>|)>>
  such that <math|\<rho\>=1> on <math|B<around*|(|0,9/10|)>> and <math|supp
  \<rho\>\<subset\>B<around*|(|0,1|)>>. For <math|x\<in\>\<bbb-R\><rsup|3>>
  let <math|\<rho\><rsub|t><around*|(|x|)>\<assign\>\<rho\><around*|(|<around*|\<langle\>|x|\<rangle\>>/t|)>>
  with <math|<around*|\<langle\>|x|\<rangle\>>\<assign\><around*|(|1+<around*|\||x|\|><rsup|2>|)><rsup|1/2>>
  and\ 

  <\equation*>
    \<sigma\><rsub|t><around*|(|x|)>\<assign\><around*|(|<frac|\<mathd\>|\<mathd\>t><around*|(|\<rho\><rsup|2><rsub|t><around*|(|x|)>|)>|)><rsup|1/2>=<around|(|-2<around*|(|<around*|\<langle\>|x|\<rangle\>>/t|)>\<rho\><around*|(|<around*|\<langle\>|x|\<rangle\>>/t|)>\<rho\><rprime|'><around*|(|<around*|\<langle\>|x|\<rangle\>>/t|)>|)><rsup|1/2>/t<rsup|1/2>.
  </equation*>

  Let <math|J<rsub|s>=><math|\<sigma\><rsub|s><around*|(|\<mathD\>|)><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1>>
  and consider the process <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>>
  defined by\ 

  <\equation>
    W<rsub|t>\<assign\><big|int><rsub|0><rsup|t>J<rsub|s>\<mathd\>X<rsub|s>=<big|sum><rsub|n\<in\>\<bbb-Z\><rsup|d>>e<rsup|i<around*|\<langle\>|n,\<cdot\>|\<rangle\>>><big|int><rsub|0><rsup|t><frac|\<sigma\><rsub|s><around*|(|n|)>|<around*|\<langle\>|n|\<rangle\>>>d
    B<rsup|n><rsub|s>,<space|2em>t\<geqslant\>0.<label|eq:def-Y>
  </equation>

  It is a centered Gaussian process with covariance\ 

  <\eqnarray*>
    <tformat|<table|<row|<cell|\<bbb-E\><around*|[|<around*|\<langle\>|W<rsub|t>,\<varphi\>|\<rangle\>><around*|\<langle\>|W<rsub|s>,\<psi\>|\<rangle\>>|]>>|<cell|=>|<cell|<big|sum><rsub|n\<in\>\<bbb-Z\><rsup|d>><frac|\<rho\><rsub|min<around*|(|s,t|)>><rsup|2><around*|(|n|)>|<around*|\<langle\>|n|\<rangle\>><rsup|2>><wide|\<varphi\>|^><around*|(|n|)><wide|<wide|\<psi\>|^><around*|(|n|)>|\<bar\>>,>>>>
  </eqnarray*>

  for any <math|\<varphi\>,\<psi\>\<in\><value|CS><around*|(|\<Lambda\>|)>>
  and <math|t,s\<geqslant\>0>, by Fubini theorem and Ito isometry. By
  dominated convergence <math|lim<rsub|t\<rightarrow\>\<infty\>>><math|\<bbb-E\><around*|[|<around*|\<langle\>|W<rsub|t>,\<varphi\>|\<rangle\>><around*|\<langle\>|W<rsub|t>,\<psi\>|\<rangle\>>|]>=<big|sum><rsub|n\<in\>\<bbb-Z\><rsup|d>><around*|\<langle\>|n|\<rangle\>><rsup|-2><wide|\<varphi\>|^><around*|(|n|)><wide|<wide|\<psi\>|^><around*|(|n|)>|\<bar\>>>
  for any <math|\<varphi\>\<nocomma\>,\<psi\>\<in\>L<rsup|2><around|(|\<Lambda\>|)>>.
  For any finite \Ptime\Q <math|T> the random field <math|W<rsub|T>> on
  <math|\<Lambda\>> has a bounded spectral support and the stopped process
  <math|W<rsup|T><rsub|t>=W<rsub|t\<wedge\>T>> for any fixed
  <math|T\<gtr\>0>, is in <math|C<around*|(|\<bbb-R\><rsub|+>,C<rsup|\<infty\>><around*|(|\<Lambda\>|)>|)>>.
  Furthermore <math|<around*|(|W<rsup|T><rsub|t>|)><rsub|t>> only depends on
  a finite subset of the Brownian motions
  <math|<around*|(|B<rsup|n>|)><rsub|n\<in\>\<bbb-Z\><rsup|d>>>. We write
  <math|g<around*|(|\<mathD\>|)>> for the Fourier multiplier operator with
  symbol <math|g>.\ 

  \;

  Observe that <math|J<rsub|t>> has the property, that for some function
  <math|f\<in\>B<rsub|p,p><rsup|s>> or <math|f\<in\>W<rsub|><rsup|s,p>> with
  <math|p\<in\><around*|[|1,\<infty\>|]>> and <math|s\<in\>\<bbb-R\>>, for
  any <math|\<alpha\>\<in\>\<bbb-R\>>

  <\equation*>
    <around*|\<\|\|\>|J<rsub|t>f|\<\|\|\>><rsub|B<rsub|p,p><rsup|s+1-\<alpha\>>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-\<alpha\>-1/2><around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,p><rsup|s>>.
  </equation*>

  \;

  We will denote by <math|<around*|\<llbracket\>|W<rsup|n><rsub|t>|\<rrbracket\>>>,
  <math|n=1,2,3>, the <math|n>-th Wick-power of the Gaussian random variable
  <math|W<rsub|t>> (under <math|\<bbb-P\>>) and introduce the convenient
  notations <math|\<bbb-W\><rsub|t><rsup|2>\<assign\>12<around*|\<llbracket\>|W<rsup|2><rsub|t>|\<rrbracket\>>>,
  <math|\<bbb-W\><rsub|t><rsup|3>\<assign\>4<around*|\<llbracket\>|W<rsub|t><rsup|3>|\<rrbracket\>>>.
  Furthermore we will write <math|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|t>|)><rsup|n>|\<rrbracket\>>,n\<in\>\<bbb-N\>>
  for the <math|n>-th Wick-power of <math|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|t>>
  . It exists for any <math|0\<less\>t\<less\>\<infty\>> and any
  <math|n\<geqslant\>1> since it is easy to see that
  <math|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|t>> has a
  covariance with a diagonal behavior which can be controlled logarithmically
  uniformly in <math|t>. These Wick powers converge as
  <math|T\<rightarrow\>\<infty\>> in spaces of distributions with
  regularities given in the following table:\ 

  <big-table|<tabular|<tformat|<cwith|1|1|1|-1|cell-tborder|0ln>|<cwith|1|1|1|-1|cell-bborder|1ln>|<cwith|2|2|1|-1|cell-tborder|1ln>|<cwith|1|1|1|1|cell-lborder|0ln>|<cwith|2|2|1|-1|cell-tsep|.5em>|<cwith|1|-1|1|-1|cell-halign|c>|<table|<row|<cell|<math|W>>|<cell|<math|\<bbb-W\><rsup|2><rsub|>>>|<cell|<math|s\<mapsto\>J<rsub|s>\<bbb-W\><rsup|3><rsub|>>>|<cell|<math|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|t>|)><rsup|n>|\<rrbracket\>>>>>|<row|<cell|<math|C<value|VV><rsup|-1/2->>>|<cell|<math|C<value|VV><rsup|-1->>>|<cell|<math|C<value|VV><rsup|-1/2->\<cap\>L<rsup|2><value|VV><rsup|-1/2->>>|<cell|<math|C<value|VV><rsup|0->>>>>>>|<label|table:reg>Regularities
  of the various stochastic objects, the domain of the time variable is
  understood to be <math|<around*|[|0,\<infty\>|]>>,
  <math|C<value|VV><rsup|\<alpha\>>=C<around*|(|<around*|[|0,\<infty\>|]>;<value|VV><rsup|\<alpha\>>|)>>
  and <math|L<rsup|2><value|VV><rsup|\<alpha\>>=L<rsup|2><around*|(|\<bbb-R\><rsub|+>;<value|VV><rsup|\<alpha\>>|)>>.
  Estimates in these norms holds a.s. and in
  <math|L<rsup|p><around*|(|\<bbb-P\>|)>> for all <math|p\<geqslant\>1> (see
  <cite|barashkov_gubinelli_variational>).>

  \ 

  We denote by <math|\<bbb-H\><rsub|a>> the space of
  <math|<around*|(|<value|CF><rsub|t>|)><rsub|t\<geqslant\>0>>-progressively
  measurable processes which are <math|\<bbb-P\>>-almost surely in
  <math|\<cal-H\>\<assign\>L<rsup|2><around*|(|\<bbb-R\><rsub|+>\<times\>\<Lambda\>|)>>.
  We say that an element <math|v> of <math|\<bbb-H\><rsub|a>> is a
  <with|font-shape|italic|drift>. Below we will need also drifts belonging to
  <math|\<cal-H\><rsup|\<alpha\>>\<assign\>L<rsup|2><around*|(|\<bbb-R\><rsub|+>;H<rsup|\<alpha\>><around*|(|\<Lambda\>|)>|)>>
  for some <math|\<alpha\>\<in\>\<bbb-R\>> where
  <math|H<rsup|\<alpha\>><around*|(|\<Lambda\>|)>> is the Sobolev space of
  regularity <math|\<alpha\>>. We denote the corresponding space with
  <math|\<bbb-H\><rsub|a><rsup|\<alpha\>>>. For any
  <math|v\<in\>\<bbb-H\><rsub|a>> define the measure <math|\<bbb-Q\><rsup|v>>
  on <math|\<Omega\>> by

  <\equation*>
    <frac|\<mathd\>\<bbb-Q\><rsup|v>|\<mathd\>\<bbb-P\>>=exp<around*|[|<big|int><rsub|0><rsup|\<infty\>>v<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsub|0><rsup|\<infty\>><around*|\<\|\|\>|v<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s|]>.
  </equation*>

  Denote with <math|\<bbb-H\><rsub|c>\<subseteq\>\<bbb-H\><rsub|a>> the set
  of drifts <math|v\<in\>\<bbb-H\><rsub|a>> for which
  <math|\<bbb-Q\><rsup|v><around*|(|\<Omega\>|)>=1>, and set
  <math|W<rsup|v>\<assign\>W-I<around*|(|v|)>>, where\ 

  <\equation*>
    I<rsub|t><around*|(|v|)>=<big|int><rsup|t><rsub|0>J<rsub|s>v<rsub|s>\<mathd\>s.
  </equation*>

  Below will need also the following operators. For all <math|t\<geqslant\>0>
  let <math|\<theta\><rsub|t>:\<bbb-R\><rsup|3>\<rightarrow\><around*|[|0,1|]>>
  be a smooth function such that\ 

  <\equation>
    <tabular|<tformat|<table|<row|<cell|\<theta\><rsub|t><around*|(|\<xi\>|)>\<sigma\><rsub|s><around*|(|\<xi\>|)>>|<cell|=>|<cell|<text|<math|0>
    for <math|s\<geqslant\>t>,>>>|<row|<cell|\<theta\><rsub|t><around*|(|\<xi\>|)>>|<cell|=>|<cell|<text|<math|1>
    for <math|<around*|\||\<xi\>|\|>\<leqslant\>t/2> \ provided that
    <math|t\<geqslant\>T<rsub|0>>>>>>>><label|theta>
  </equation>

  for some <math|T<rsub|0>\<gtr\>0>. For example one can fix smooth functions
  <math|<wide|\<theta\>|~>,\<eta\>:\<bbb-R\><rsup|3>\<rightarrow\>\<bbb-R\><rsub|+>>
  such that <math|<wide|\<theta\>|~><around*|(|\<xi\>|)>=1> if
  <math|<around*|\||\<xi\>|\|>\<leqslant\>1/2> and
  <math|<wide|\<theta\>|~><around*|(|\<xi\>|)>=0> if
  <math|<around*|\||\<xi\>|\|>\<geqslant\>2/3> ,
  <math|\<eta\><around*|(|\<xi\>|)>=1> if
  <math|<around*|\||\<xi\>|\|>\<leqslant\>1> and
  <math|\<eta\><around*|(|\<xi\>|)>=0> if
  <math|<around*|\||\<xi\>|\|>\<geqslant\>2>. Then let
  <math|<wide|\<theta\>|~><rsub|t><around*|(|\<xi\>|)>\<assign\><wide|\<theta\>|~><around*|(|\<xi\>/t|)>>
  and define <math|>

  <\equation*>
    \<theta\><rsub|t><around*|(|\<xi\>|)>=<around*|(|1-\<eta\><around*|(|\<xi\>|)>|)><wide|\<theta\>|~><rsub|t><around*|(|\<xi\>|)>+\<zeta\><around*|(|t|)>\<eta\><around*|(|\<xi\>|)><wide|\<theta\>|~><rsub|t><around*|(|\<xi\>|)>
  </equation*>

  where <math|><math|\<zeta\><around*|(|t|)>:\<bbb-R\><rsub|+>\<rightarrow\>\<bbb-R\>>
  is a smooth function such that <math|\<zeta\><around*|(|t|)>=0> for
  <math|t\<leqslant\>10> and <math|\<zeta\><around*|(|t|)>=1> for
  <math|t\<geqslant\>3>. Then eq<nbsp><eqref|theta> will hold with
  <math|T<rsub|0>=3>. We will let

  <\equation>
    f<rsup|\<flat\>>\<assign\>\<theta\><around*|(|\<mathD\>|)>f<label|eq:flat>
  </equation>

  for any <math|f\<in\><value|CS><rprime|'><around*|(|\<Lambda\>|)>>.

  \;

  Our aim here to study the measures <math|\<mu\><rsub|T>> defined on
  <math|<value|VV><rsup|-1/2-\<varepsilon\>>> as\ 

  <\equation*>
    <frac|\<mathd\>\<mu\><rsub|T>|\<mathd\>\<bbb-P\>>=e<rsup|-V<rsub|T><around*|(|W<rsub|T>|)>>
  </equation*>

  for <math|\<varphi\>\<in\>C<rsup|\<infty\>><around*|(|\<Lambda\>|)>>

  <\equation>
    V<rsub|T><around*|(|\<varphi\>|)>\<assign\>\<lambda\><big|int><rsub|\<Lambda\>><around*|(|\<varphi\><rsup|4>-a<rsub|T>\<varphi\><rsup|2>+b<rsub|T>|)>\<mathd\>x,<label|eq:def-V>
  </equation>

  with suitable <math|a<rsub|T>,b<rsub|T>\<rightarrow\>\<infty\>>. For
  convenience the measure <math|\<mu\><rsup|T>> is not normalized and, wrt.
  to the notations in the introduction we have

  <\equation*>
    <frac|\<mathd\>\<bbb-P\><rsup|T>|\<mathd\>\<mu\><rsup|T>>=<frac|1|\<mu\><rsup|T><around*|(|\<Omega\>|)>>.
  </equation*>

  \;

  With these notations we can recall the following results of
  <cite|barashkov_gubinelli_variational>.

  <\theorem>
    <label|eq:th1>For any <math|a<rsub|T>,b<rsub|T>\<in\>\<bbb-R\>>, and
    \ <math|f:<value|VV><rsup|-1/2-\<varepsilon\>><around*|(|\<Lambda\>|)>\<rightarrow\>\<bbb-R\>>
    with linear growth, recall<nbsp><eqref|eq:def-V> and let
    <math|V<rsup|f><rsub|T><around*|(|\<varphi\>|)>\<assign\>f<around*|(|\<varphi\>|)>+V<rsub|T><around*|(|\<varphi\>|)>.>
    Then the formula\ 

    <\equation>
      <big|int><rsub|<value|CS><rprime|'><around*|(|\<Lambda\>|)>>e<rsup|-V<rsup|f><rsub|T><around*|(|\<varphi\>|)>>\<mu\><around*|(|\<mathd\>\<varphi\>|)><rsup|>=-log
      \<bbb-E\><around*|[|e<rsup|-V<rsup|f><rsub|T><around*|(|W<rsub|T>|)>><rsup|>|]>=inf<rsub|u\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|V<rsup|f><rsub|T><around*|(|W<rsub|T>+I<rsub|T><around*|(|u|)>|)>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2><around*|(|\<Lambda\>|)>>\<mathd\>t|]><label|eq:contr-prob>
    </equation>

    holds for any finite <math|T>.
  </theorem>

  This is a consequence of the more general Boué\UDupuis formula which can be
  stated as follows.

  <\theorem>
    <dueto|BD formula>Assume <math|F :C<around*|(|<around*|[|0,T|]>,C<rsup|\<infty\>><around*|(|\<Lambda\>|)>|)>\<rightarrow\>\<bbb-R\>>,
    be Borel measurable and such that there exist
    <math|p,q\<in\><around*|(|1,\<infty\>|)>>, with <math|1/p+1/q=1>,
    <math|\<bbb-E\><around*|[|<around*|\||F<around*|(|W|)>|\|><rsup|p>|]>\<less\>\<infty\>>
    and <math|\<bbb-E\><around*|[|<around*|\||e<rsup|-F<around*|(|W|)>>|\|><rsup|q>|]>\<less\>\<infty\>>(where
    we can regard <math|W> as an element of
    <math|C<around*|(|<around*|[|0,T|]>,C<rsup|\<infty\>><around*|(|\<Lambda\>|)>|)>>
    by restricting to <math|<around*|[|0,T|]>>). Then

    <\equation>
      -log \<bbb-E\><around*|[|e<rsup|-F<around*|(|W|)>>|]>=inf<rsub|u\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|F<around*|(|W+I<around*|(|u|)>|)>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsub|L<rsup|2><around*|(|\<Lambda\>|)>><rsup|2>|]>.<label|eq:bd>
    </equation>
  </theorem>

  We will use several times below the Boué\UDupuis formula<nbsp><eqref|eq:bd>
  in order to control exponential integrability of various functional. By a
  suitable choice of renormalization and a change of variables in the control
  problem<nbsp><eqref|eq:contr-prob> we were able
  in<nbsp><cite|barashkov_gubinelli_variational> to control the functional in
  Theorem<nbsp><reference|eq:th1> uniformly up to infinity:

  <\theorem>
    <label|thm:equicoerc>There exist a sequence
    <math|<around*|(|a<rsub|T>,b<rsub|T>|)><rsub|T>> with
    <math|a<rsub|T>,b<rsub|T>\<rightarrow\>\<infty\>> as
    <math|T\<rightarrow\>\<infty\>>, such that\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|\<bbb-E\><around*|[|V<rsup|f><rsub|T><around*|(|W<rsub|T>+I<rsub|T><around*|(|u|)>|)>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|t>|\<\|\|\>><rsub|L<rsup|2><around*|(|\<Lambda\>|)>>\<mathd\>t|]>>>|<row|<cell|>|<cell|=>|<cell|\<bbb-E\><around*|[|\<Psi\><rsup|f><rsub|T><around*|(|W,I<rsub|T><around*|(|u|)>|)>+\<lambda\><big|int><around|(|I<rsub|T><around*|(|u|)>|)><rsup|4>+<frac|1|2><around*|\<\|\|\>|l<rsup|T><around*|(|u|)>|\<\|\|\>><rsub|\<cal-H\>><rsup|2>|]>>>>>
    </eqnarray*>

    where\ 

    <\equation>
      l<rsub|t><rsup|T><around*|(|u|)>\<assign\>u<rsub|t>+\<lambda\>\<bbb-1\><rsub|t\<leqslant\>T>\<bbb-W\><rsup|<around*|\<langle\>|3|\<rangle\>>><rsub|t>+\<lambda\>\<bbb-1\><rsub|t\<leqslant\>T>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|2>\<succ\>I<rsub|t><rsup|\<flat\>><around*|(|u|)>|)><label|eq:renorm-drift>
    </equation>

    and the functional <math|\<Psi\><rsup|f><rsub|T>> satisfies the following
    bound

    <\equation*>
      <around*|\||\<Psi\><rsup|f><rsub|T><around*|(|W,I<rsub|T><around*|(|u|)>|)>|\|>\<leqslant\>Q<rsub|T><around*|(|\<bbb-W\>|)>+<frac|1|4><around*|(|<around*|\<\|\|\>|I<rsub|T><around*|(|u|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<around*|\<\|\|\>|l<rsup|T><around*|(|u|)>|\<\|\|\>><rsub|\<cal-H\>><rsup|2>|)>
    </equation*>

    where <math|Q<rsub|T><around*|(|W|)>> is a function of <math|W>
    independent of <math|u> and such that <math|sup<rsub|T>
    \<bbb-E\><around*|[|<around*|\||Q<rsub|T><around*|(|W|)>|\|>|]>\<less\>\<infty\>>.
  </theorem>

  Recall that <math|I<rsub|t><rsup|\<flat\>><around*|(|u|)>=\<theta\><around*|(|\<mathD\>|)>I<rsub|t><around*|(|u|)>>
  by<nbsp><eqref|eq:flat>.

  <section|Construction of the drift measure>

  We start now to implement the strategy discussed in the introduction:
  construct a shifted measure sufficiently similar to
  <math|\<Phi\><rsup|4><rsub|3>>. Intuitively the
  <math|\<Phi\><rsup|4><rsub|3>> measure should give rise to a canonical
  process which is a shift of the Gaussian free field with a drift of the
  form given by eq.<nbsp><eqref|eq:renorm-drift>. Indeed this drift <math|u>
  should be the optimal drift in the variational formula.

  \;

  A small twist is given by the fact that the relevant Gaussian free field
  entering these considerations is not the process <math|W=W<around*|(|X|)>>
  but that obtained from the shifted canonical process
  <math|X<rsup|u><rsub|t>=X<rsub|t>-<big|int><rsub|0><rsup|t>u<rsub|s>\<mathd\>s>
  which we denote by

  <\equation*>
    W<rsup|u>\<assign\>W<around*|(|X<rsup|u>|)>=W-I<around*|(|u|)>.
  </equation*>

  Moreover for technical reasons we have to modify the drift in large scales
  and add some coercive term which will allow later to prove some useful
  estimates. We define the functional

  <\equation>
    \<Xi\><rsub|s><around*|(|W,u|)>\<assign\>-\<lambda\>J<rsub|s>\<bbb-W\><rsub|s><rsup|3>-\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|\<bbb-W\><rsub|s><rsup|2>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|u|)>|)>-J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|(|<around*|\<llbracket\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|s>|\<rrbracket\>><rsup|n>|)>,<space|2em>s\<geqslant\>0,<label|eq:pre-pre-drift>
  </equation>

  where <math|<wide|T|\<bar\>>\<gtr\>0> is a constant which will be fixed
  later and where we understand all the Wick renormalizations to be given
  functions of <math|W>. We look now for the solution <math|u> of the
  equation

  <\equation>
    u=\<Xi\><around*|(|W<rsup|u>,u|)>=\<Xi\><around*|(|W-I<around*|(|u|)>,u|)>.<label|eq:pre-drift>
  </equation>

  Expanding the Wick polynomials appearing in
  <math|\<Xi\><around*|(|W-I<around*|(|u|)>,u|)>> we obtain the equation

  <\equation>
    <tabular|<tformat|<cwith|1|-1|3|3|cell-hyphen|t>|<table|<row|<cell|u<rsub|s>>|<cell|=>|<\cell>
      \<Xi\><around*|(|W-I<around*|(|u|)>,u|)>
    </cell>>|<row|<cell|>|<cell|=>|<\cell>
      -\<lambda\>J<rsub|s><around*|[|\<bbb-W\><rsub|s><rsup|3>-\<bbb-W\><rsub|s><rsup|2>I<rsub|s><around*|(|u|)>+12W<rsub|s><around*|(|I<rsub|s><around*|(|u|)>|)><rsup|2>-4<around*|(|I<rsub|s><around*|(|u|)>|)><rsup|3>|]>
    </cell>>|<row|<cell|>|<cell|>|<\cell>
      -\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>J<rsub|s><around*|[|<around*|(|<around*|(|\<bbb-W\><rsub|s><rsup|2>-24W<rsub|s>I<rsub|s><around*|(|u|)>+12<around*|(|I<rsub|s><around*|(|u|)>|)><rsup|2>|)>|)>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|u|)>|]>
    </cell>>|<row|<cell|>|<cell|>|<\cell>
      -<big|sum><rsub|i=0><rsup|n><choose|n|i>J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|[|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|s>|)><rsup|i>|\<rrbracket\>><around*|(|-<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|s><around*|(|u|)>|)><rsup|n-i>|]>
    </cell>>>>><label|eq:drift>
  </equation>

  for all <math|s\<geqslant\>0>. This is an integral equation for
  <math|t\<mapsto\>u<rsub|t>> with smooth coefficients depending smoothly on
  <math|W> and can be solved via standard methods. Since the coefficients are
  of polynomial growth we must expect explosion in finite time, so we have to
  be careful. Note that for any finite time the process
  <math|<around*|(|u<rsub|s>|)><rsub|s\<gtr\>0>> has bounded spectral
  support. As a consequence we can solve the equation in <math|L<rsup|2>> and
  as long as <math|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|u|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>
  is finite we can see from the equation that
  <math|sup<rsub|s\<leqslant\>t><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>>
  is finite. Therefore by the existence of local solutions we have that, for
  all <math|N\<geqslant\>0>, the stopping time

  <\equation*>
    \<tau\><rsub|N>\<assign\>inf<around*|{|t\<geqslant\>0<mid|\|><big|int><rsup|t><rsub|0><around*|\<\|\|\>|u|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s\<geqslant\>N|}>,
  </equation*>

  is strictly positive <math|\<bbb-P\>>-almost surely and <math|u> exists up
  to the (explosion) time <math|T<rsub|exp>\<assign\>sup<rsub|N\<in\>\<bbb-N\>>
  \<tau\><rsub|N>>. Moreover, by construction, the process
  <math|u<rsub|t><rsup|N>\<assign\><with|math-font|Bbb*|1><rsub|<around*|{|t\<leqslant\>\<tau\><rsub|N>|}>>u<rsub|t>>
  satisfies Novikov's condition, so it is in <math|\<bbb-H\><rsub|c>> and by
  Girsanov transformation we can define the probability measure on
  <math|C<around*|(|\<bbb-R\><rsub|+>,<value|VV><rsup|-1/2-\<varepsilon\>><around*|(|\<Lambda\>|)>|)>>
  given by

  <\equation*>
    \<mathd\>\<bbb-Q\><rsup|u<rsup|N>>\<assign\>e<rsup|<big|int><rsup|\<infty\>><rsub|0>u<rsub|s><rsup|N>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|\<infty\>><rsub|0><around*|\<\|\|\>|u<rsub|s><rsup|N>|\<\|\|\>><rsup|2><rsub|L<rsup|2><around*|(|\<Lambda\>|)>>\<mathd\>s>\<mathd\>\<bbb-P\>,
  </equation*>

  and under which <math|X<rsup|u<rsup|N>><rsub|t>=X<rsub|t>-<big|int><rsup|t><rsub|0>u<rsub|s><rsup|N>\<mathd\>s>
  is a cylindrical Brownian motion. In particular
  <math|<around*|(|W<rsup|u<rsup|N>><rsub|t>|)><rsub|t\<geqslant\>0>>, given
  by <math|<big|int><rsup|t><rsub|0>J<rsub|s>\<mathd\>X<rsup|u<rsup|N>><rsub|s>>
  has under <math|\<bbb-Q\><rsup|u<rsup|N>>> the same law as
  <math|<around*|(|W<rsub|t>|)><rsub|t\<geqslant\>0>> has under
  <math|\<bbb-P\>>. Moreover we have that
  <math|W<rsup|u<rsup|N>><rsub|s>=W<rsup|u><rsub|s>> for
  <math|0\<leqslant\>s\<leqslant\>\<tau\><rsub|N>> and that <math|u>
  satisfies the equation

  <\equation>
    u<rsub|s>=-\<lambda\>J<rsub|s>\<bbb-W\><rsub|s><rsup|u,3>-\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|\<bbb-W\><rsub|s><rsup|u,2>\<succ\>I<rsub|t><rsup|\<flat\>><around*|(|u|)>|)>-J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|(|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|s>|)><rsup|n>|\<rrbracket\>>|)>,<label|eq:drift-uu><space|2em>s\<in\><around*|[|0,\<tau\><rsub|N>|]>,
  </equation>

  where we introduced the notations <math|\<bbb-W\><rsub|s><rsup|u,3>\<assign\>4<around*|\<llbracket\>|<around*|(|W<rsub|s><rsup|u>|)><rsup|3>|\<rrbracket\>>>
  and <math|\<bbb-W\><rsub|s><rsup|u,2>\<assign\>12<around*|\<llbracket\>|<around*|(|W<rsup|u><rsub|s>|)><rsup|2>|\<rrbracket\>>>.

  \;

  Note that here the Wick powers are still taken to be given functions of
  <math|W>, i.e we are still taking the Wick ordering with respect to the law
  of <math|W> under <math|\<bbb-P\>> (or the law of <math|W<rsup|u<rsup|N>>>
  under <math|\<bbb-Q\><rsup|u<rsup|N>>>).

  \;

  If we think to the terms containing <math|W<rsup|u>> as given (that is, we
  ignore their dependence on <math|u>), eq.<nbsp><eqref|eq:drift-uu> is a
  linear integral equation in <math|u> which can be estimated via
  Gronwall-type arguments. In order to do so, let us denote by
  <math|U:H\<mapsto\><wide|u|^>> the solution map of the equation

  <\equation>
    <wide|u|^>=\<Xi\><around*|(|H,<wide|u|^>|)>.<label|eq:map-U>
  </equation>

  This last equation is linear and therefore has nice global solutions (let's
  say in <math|C<around*|(|\<bbb-R\><rsub|+>,L<rsup|2>|)>>) and by uniqueness
  and eq.<nbsp><eqref|eq:drift-uu> we have
  <math|u<rsub|t>=U<rsub|t><around*|(|W<rsup|u>|)>> for
  <math|t\<in\><around*|[|0,T<rsub|exp>|)>>. From this perspective the
  residual dependence on <math|u> will not play any role since under the
  shifted measure the law of the process <math|W<rsup|u>> does not depend on
  <math|u>. By standard paraproduct estimates we have

  <\eqnarray*>
    <tformat|<table|<row|<cell|<around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsub|L<rsup|\<infty\>>>>|<cell|\<lesssim\>>|<cell|<wide|H|~><rsub|t>+<big|int><rsup|t><rsub|0><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>><around*|\<\|\|\>|J<rsup|2><rsub|s><around*|(|\<bbb-W\><rsub|s><rsup|u,2>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|u|)>|)>|\<\|\|\>><rsub|L<rsup|\<infty\>>>\<mathd\>s>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<wide|H|~><rsub|t>+<rsub|><wide|T|\<bar\>><rsup|-\<varepsilon\>><big|int><rsup|t><rsub|0><around*|\<langle\>|s|\<rangle\>><rsup|-3/2><around*|\<\|\|\>|\<bbb-W\><rsub|s><rsup|u,2>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<varepsilon\>>><around*|\<\|\|\>|I<rsub|s><rsup|\<flat\>><around*|(|u|)>|\<\|\|\>><rsub|L<rsup|\<infty\>>>\<mathd\>s,>>>>
  </eqnarray*>

  where we have used the presence of the cutoff
  <math|<with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>>
  to introduce the small factor <math|T<rsup|-\<varepsilon\>>> and we have
  employed the notation\ 

  <\equation*>
    <wide|H|~><rsub|t>=<big|int><rsup|t><rsub|0><around*|[|<around|\<\|\|\>|J<rsup|2><rsub|s>\<bbb-W\><rsup|u<rsup|N>,3><rsub|s><rsup|>|\<\|\|\>><rsub|L<rsup|\<infty\>>>+<around*|\<\|\|\>|J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|(|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|s>|)><rsup|n>|\<rrbracket\>>|)>|\<\|\|\>><rsub|L<rsup|\<infty\>>>|]>\<mathd\>s
  </equation*>

  <\equation*>
    \<lesssim\><big|int><rsup|t><rsub|0><frac|1|<around*|\<langle\>|s|\<rangle\>><rsup|1/2-\<varepsilon\>>><around|\<\|\|\>|J<rsub|s>\<bbb-W\><rsup|u<rsup|N>,3><rsub|s><rsup|>|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<mathd\>s+<big|int><rsup|t><rsub|0><frac|1|<around*|\<langle\>|s|\<rangle\>><rsup|3/2>><around|\<\|\|\>|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|s>|)><rsup|n>|\<rrbracket\>>|\<\|\|\>><rsub|H<rsup|-1/2>>\<mathd\>s.
  </equation*>

  Therefore, by Gronwall's lemma

  <\equation>
    <tabular|<tformat|<table|<row|<cell|sup<rsub|t\<leqslant\>\<tau\><rsub|N>><around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsub|L<rsup|\<infty\>>>>|<cell|\<lesssim\>>|<cell|<wide|H|~><rsub|\<tau\><rsub|N>>*exp<around*|(|C
    <wide|T|\<bar\>><rsup|-\<varepsilon\>><big|int><rsup|\<tau\><rsub|N>><rsub|0><around|\<\|\|\>|\<bbb-W\><rsub|s><rsup|u,2>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|><frac|\<mathd\>s|<around*|\<langle\>|s|\<rangle\>><rsup|1+\<varepsilon\>>>|)>.>>>>><label|boundIu>
  </equation>

  Under <math|\<bbb-Q\><rsup|u<rsup|N>>>, the terms in
  <math|<wide|H|~><rsub|\<tau\><rsub|N>>> are in all the <math|L<rsup|p>>
  spaces by hypercontractivity and moreover for any <math|p\<geqslant\>1> one
  can choose <math|<wide|T|\<bar\>>> large enough to that also the
  exponential term is in <math|L<rsup|p>>. Using eq.<nbsp><eqref|eq:drift-uu>
  it is then not difficult to show that <math|\<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N<rsub|1>>>><around*|[|<around*|\<\|\|\>|u<rsup|N<rsub|2>>|\<\|\|\>><rsup|p><rsub|<with|math-font|cal|H><rsup|-1/2-\<varepsilon\>>>|]>\<less\>\<infty\>>
  for any <math|p\<gtr\>1> (again provided we take <math|<wide|T|\<bar\>>>
  large enough depending on <math|p>) as long as
  <math|N<rsub|1>\<gtr\>N<rsub|2>>. By the spectral properties of <math|J>
  and the equation for <math|u>, the process
  <math|t\<mapsto\><with|math-font|Bbb*|1><rsub|<around*|{|t\<leqslant\>T|}>>u<rsub|t>>
  is spectrally supported in a ball of radius <math|T>, so we get in
  particular that

  <\equation*>
    \<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N<rsub|1>>>><around*|[|<big|int><rsup|\<tau\><rsub|N<rsub|2>>\<wedge\>T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s|]>\<lesssim\>T<rsup|1+\<varepsilon\>>,
  </equation*>

  uniformly for any choice of <math|N<rsub|1>\<geqslant\>N<rsub|2>\<geqslant\>0>.

  <\lemma>
    <label|Lemma:changeofvariables>The family
    <math|<around*|(|\<bbb-Q\><rsup|u<rsup|N>>|)><rsub|N>> weakly converges
    to a limit <math|\<bbb-Q\><rsup|u>> on
    <math|C<around*|(|\<bbb-R\><rsub|+>,<value|VV><rsup|-3/2-\<varepsilon\>>|)>>.
    Under <math|\<bbb-Q\><rsup|u>> it holds <math|T<rsub|exp>=\<infty\>>
    almost surely and <math|Law<rsub|\<bbb-Q\><rsup|u>><around*|(|X<rsup|u>|)>=Law<rsub|\<bbb-P\>><around*|(|X|)>>.
    Moreover for any finite <math|T>

    <\equation*>
      <frac|\<mathd\>\<bbb-Q\><rsup|u>\|<rsub|<value|CF><rsub|T>>|\<mathd\>\<bbb-P\>\|<rsub|<value|CF><rsub|T>>>=exp<around*|(|<big|int><rsup|T><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s|)>.
    </equation*>
  </lemma>

  <\proof>
    Consider the filtration <math|<around*|(|<CG><rsub|N>=<value|CF><rsub|\<tau\><rsub|N>>|)><rsub|N>>
    and observe that <math|<around*|(|<around*|\<nobracket\>|\<bbb-Q\><rsup|u<rsup|N>>|\|><rsub|<value|CG><rsub|N>>|)><rsub|N>>
    is a consistent family of inner regular probability distributions and
    therefore there exists a unique extension <math|\<bbb-Q\><rsup|u>> to
    <math|<value|CG><rsub|\<infty\>>=\<vee\><rsub|N><value|CG><rsub|N>>. Next
    observe that <math|<around*|{|T<rsub|exp>\<less\>\<infty\>|}>=<big|cup><rsub|T\<in\>\<bbb-N\>><around*|{|T<rsub|exp>\<less\>T|}>\<subset\><big|cup><rsub|T\<in\>\<bbb-N\>><big|cap><rsub|N\<in\>\<bbb-N\>><around*|{|\<tau\><rsub|N>\<less\>T|}>>
    and that for any <math|N,T\<less\>\<infty\>>, we have

    <\equation*>
      \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<big|int><rsup|\<tau\><rsub|N>\<wedge\>T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s|]>=\<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N>>><around*|[|<big|int><rsup|\<tau\><rsub|N>\<wedge\>T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s|]>\<lesssim\>T<rsup|1+\<varepsilon\>>.
    </equation*>

    On the event <math|<around*|{|\<tau\><rsub|N>\<leqslant\>T|}>> we have

    <\equation*>
      <big|int><rsup|\<tau\><rsub|N>\<wedge\>T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s=N,
    </equation*>

    and therefore we also have \ <math|\<bbb-Q\><rsup|u><around*|(|<around*|{|\<tau\><rsub|N>\<leqslant\>T|}>|)>\<leqslant\>C
    T<rsup|1+\<varepsilon\>>N<rsup|-1>> which in turn implies
    <math|\<bbb-Q\><rsup|u><around*|(|T<rsub|exp>\<less\>T|)>=0>. This proves
    that <math|T<rsub|exp>=+\<infty\>> under <math|\<bbb-Q\><rsup|u>>, almost
    surely. As a consequence we can extend <math|\<bbb-Q\><rsup|u>> to all of
    <math|<value|CF>=\<vee\><rsub|T><value|CF><rsub|T>> since for any
    <math|A\<in\><value|CF><rsub|T>> we can set\ 

    <\equation*>
      \<bbb-Q\><rsup|u><around*|(|A|)>=\<bbb-Q\><rsup|u><around*|(|A\<cap\><around*|{|T<rsub|exp>=+\<infty\>|}>|)>=lim<rsub|N>
      \<bbb-Q\><rsup|u><around*|(|A\<cap\><around*|{|T<rsub|exp>=+\<infty\>,\<tau\><rsub|N>\<geqslant\>T|}>|)>=lim<rsub|N>
      \<bbb-Q\><rsup|u<rsup|N>><around*|(|A\<cap\><around*|{|\<tau\><rsub|N>\<geqslant\>T|}>|)>.
    </equation*>

    If <math|A\<in\><value|CF><rsub|T>> then

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|A><around*|(|X<rsup|u>|)>|]>>|<cell|=>|<cell|lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>><around*|(|X<rsup|u>|)>|]>=lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N>>><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>><around*|(|X<rsup|u<rsup|N>>|)>|]>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-P\>><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>><around*|(|X|)>|]>>>>>>
    </equation*>

    and

    <\equation*>
      lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-P\>><around*|[|<with|math-font|Bbb*|1><rsub|<around*|{|T\<gtr\>\<tau\><rsub|N>|}>><around*|(|X|)>|]>=lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N>>><around*|[|<with|math-font|Bbb*|1><rsub|<around*|{|T\<gtr\>\<tau\><rsub|N>|}>><around*|(|X<rsup|u<rsup|N>>|)>|]>=lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|<around*|{|T\<gtr\>\<tau\><rsub|N>|}>><around*|(|X<rsup|u>|)>|]>\<rightarrow\>0.
    </equation*>

    This establishes that <math|Law<rsub|\<bbb-Q\><rsup|u>><around*|(|X<rsup|u>|)>=Law<rsub|\<bbb-P\>><around*|(|X|)>>.
    On the other hand if <math|A\<in\><value|CF><rsub|T>> we have, using the
    martingale property of the Girsanov density,

    <\equation*>
      \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|A>|]>=lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>>|]>=lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N>>><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>>|]>
    </equation*>

    <\equation*>
      =lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>>e<rsup|<big|int><rsup|\<tau\><rsub|N>><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|\<tau\><rsub|N>><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>|]>
    </equation*>

    <\equation*>
      =lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<leqslant\>\<tau\><rsub|N>|}>>e<rsup|<big|int><rsup|T><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>|]>.
    </equation*>

    And also

    <\equation*>
      lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><around*|[|<with|math-font|Bbb*|1><rsub|A\<cap\><around*|{|T\<gtr\>\<tau\><rsub|N>|}>>e<rsup|<big|int><rsup|T><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>|]>\<leqslant\>lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><around*|[|<with|math-font|Bbb*|1><rsub|<around*|{|T\<gtr\>\<tau\><rsub|N>|}>>e<rsup|<big|int><rsup|T><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>|]>
    </equation*>

    <\equation*>
      =lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><around*|[|<with|math-font|Bbb*|1><rsub|<around*|{|T\<gtr\>\<tau\><rsub|N>|}>>e<rsup|<big|int><rsup|\<tau\><rsub|N>><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|\<tau\><rsub|N>><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>|]>=lim<rsub|N\<rightarrow\>\<infty\>>\<bbb-E\><rsub|\<bbb-Q\><rsup|u<rsup|N>>><around*|[|<with|math-font|Bbb*|1><rsub|<around*|{|T\<gtr\>\<tau\><rsub|N>|}>>|]>=0.
    </equation*>

    As a consequence

    <\equation*>
      \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|A>|]>=\<bbb-E\><around*|[|<with|math-font|Bbb*|1><rsub|A>e<rsup|<big|int><rsup|T><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>|]>
    </equation*>

    and therefore

    <\equation*>
      <frac|\<mathd\>\<bbb-Q\><rsup|u>\|<rsub|<value|CF><rsub|T>>|\<mathd\>\<bbb-P\>\|<rsub|<value|CF><rsub|T>>>=e<rsup|<big|int><rsup|T><rsub|0>u<rsub|s>\<mathd\>X<rsub|s>-<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|s>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>s>,
    </equation*>

    as claimed.
  </proof>

  The following lemma will also be useful in the sequel and it is a
  consequence of the above discussion:\ 

  <\lemma>
    For any <math|p\<gtr\>1> there exists a suitable choice of
    <math|<wide|T|\<bar\>>> such that\ 

    <\equation*>
      \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|sup<rsub|t\<geqslant\>0><around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsup|p><rsub|L<rsup|\<infty\>>>|]>\<less\>\<infty\>.
    </equation*>
  </lemma>

  <\proof>
    This follows from the bound<nbsp><eqref|boundIu>, after choosing
    <math|<wide|T|\<bar\>>> large enough.\ 
  </proof>

  <subsection|Proof of absolute continuity>

  In this section we prove that the measure <math|\<mu\><rsub|T>> is
  absolutely continuous with respect to the measure <math|\<bbb-Q\><rsup|u>>
  we constructed in Lemma<nbsp><reference|Lemma:changeofvariables>. First
  recall that the measures <math|\<mu\><rsub|T>> defined on <math|\<Omega\>>
  as\ 

  <\equation*>
    <frac|\<mathd\>\<mu\><rsub|T>|\<mathd\>\<bbb-P\>>=e<rsup|-V<rsub|T><around*|(|W<rsub|T>|)>>
  </equation*>

  can be described, using Lemma<nbsp><reference|Lemma:changeofvariables>, as
  a perturbation of <math|\<bbb-Q\><rsup|u>> with density <math|D<rsub|T>>
  given by

  <\equation*>
    D<rsub|T>\<assign\><around*|\<nobracket\>|<frac|\<mathd\>\<mu\><rsub|T>|\<mathd\>\<bbb-Q\><rsup|u>>|\|><rsub|<value|CF><rsub|T>>=<around*|\<nobracket\>|<frac|\<mathd\>\<mu\><rsub|T>|\<mathd\>\<bbb-P\>>|\|><rsub|<value|CF><rsub|T>><around*|\<nobracket\>|<frac|\<mathd\>\<bbb-P\>|\<mathd\>\<bbb-Q\><rsup|u>>|\|><rsub|<value|CF><rsub|T>>=e<rsup|-V<rsub|T><around*|(|W<rsub|T>|)>-<big|int><rsup|T><rsub|0>u\<mathd\>X+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|u<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t>,
  </equation*>

  at least on <math|<value|CF><rsub|T>>.

  <\lemma>
    <label|lemma:localestimate>There exists a <math|p\<gtr\>1>, such that for
    any <math|K\<gtr\>0>,

    <\equation*>
      sup<rsub|T>\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|\||D<rsub|T>|\|><rsup|p><with|math-font|Bbb*|1><rsub|<around*|{|<around*|\<\|\|\>|W<rsub|\<infty\>>|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<leqslant\>K|}>>|]>\<less\>\<infty\>.
    </equation*>

    in particular, the family <math|<around*|(|D<rsub|T>|)><rsub|T>> is
    uniformly integrable under <math|\<bbb-Q\><rsup|u>>.
  </lemma>

  The proof of this claim is given in Section<nbsp><reference|sec:lp-bounds>
  below.

  <\corollary>
    The family of measures <math|<around*|(|\<mu\><rsub|T>|)><rsub|T\<geqslant\>0>>
    is sequentially compact w.r.t. strong convergence on
    <math|><math|<around*|(|\<Omega\>,<value|CF>|)>>. Furthermore any
    accumulation point is absolutely continuous with respect to
    <math|\<bbb-Q\><rsup|u>>.
  </corollary>

  <\proof>
    We choose a subsequence (not relabeled) such that
    <math|D<rsub|T>\<rightarrow\>D<rsub|\<infty\>>> weakly in
    <math|L<rsup|1><around*|(|\<bbb-Q\><rsup|u>|)>>, for some
    <math|D<rsub|\<infty\>>\<in\>L<rsup|1><around*|(|\<bbb-Q\><rsup|u>|)>>.
    It always exists by uniform integrability. We now claim that for any
    <math|A\<in\><value|CF>>\ 

    <\equation*>
      lim<rsub|T\<rightarrow\>\<infty\>>\<mu\><rsub|T><around*|(|A|)>=<big|int><rsub|A>D<rsub|\<infty\>>\<mathd\>\<bbb-Q\><rsup|u><rsub|>.
    </equation*>

    It is enough to check this for <math|A\<in\><value|CF><rsub|S>> for any
    <math|S\<in\>\<bbb-R\><rsub|+>> since these generate <math|<value|CF>>.
    But there we have for <math|T\<geqslant\>S>,

    <\equation*>
      \<mu\><rsub|T><around*|(|A|)>=<big|int><rsub|A>D<rsub|T>\<mathd\>\<bbb-Q\><rsup|u><rsub|>\<rightarrow\><big|int><rsub|A>D<rsub|\<infty\>>\<mathd\>\<bbb-Q\><rsup|u><rsub|>
    </equation*>

    by weak <math|L<rsup|1>> convergence. \ 
  </proof>

  Recall that the <math|\<Phi\><rsub|3><rsup|4>> measure can be defined as a
  weak limit of the measures <math|<wide|\<mu\>|~><rsub|T>> on
  <math|<value|VV><rsup|-1/2-\<varepsilon\>>> given by\ 

  <\equation*>
    <big|int>f <around*|(|\<varphi\>|)><wide|\<mu\>|~><rsub|T><around*|(|\<mathd\>\<varphi\>|)>=<big|int>f<around*|(|\<varphi\>|)>e<rsup|-V<rsub|T><around*|(|\<varphi\>|)>>\<theta\><rsub|T><around*|(|\<mathd\>\<varphi\>|)>=\<bbb-E\><rsub|\<bbb-P\>><around*|[|f<around*|(|W<rsub|T>|)>e<rsup|-V<rsub|T><around*|(|W<rsub|T>|)>>|]>
  </equation*>

  where <math|\<theta\><rsub|T>> is the gaussian measure with covariance
  <math|\<rho\><rsup|2><rsub|T><around*|(|\<mathD\>|)><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-2>>.
  From this together with the above considerations we see that any
  accumulation point <math|<wide|\<mu\>|~><rsub|\<infty\>>> of
  <math|<wide|\<mu\>|~><rsub|T>> satisfies\ 

  <\equation>
    <wide|\<mu\>|~><rsub|\<infty\>><around*|(|A|)>=\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<with|math-font|Bbb*|1><rsub|A><around*|(|W<rsub|\<infty\>>|)>D<rsub|\<infty\>>|]><label|eq:Phi4def>
  </equation>

  for some <math|D<rsub|\<infty\>>\<in\>L<rsup|1><around*|(|\<bbb-Q\><rsup|u>|)>>.

  <subsection|<math|L<rsup|p>> bounds><label|sec:lp-bounds>

  Now we will prove local <math|L<rsup|p>>-bounds on the density
  <math|D<rsub|T>>. In the sequel we will denote <math|<wide|W|~>=W<rsup|u>>,
  with <math|u> satisfying <eqref|eq:drift>, namely
  <math|u=U<around*|(|<wide|W|~>|)>>. Before we proceed let us study how the
  functional <math|U<around*|(|<wide|W|~>|)>> behaves under shifts of
  <math|<wide|W|~>>, since later we will want to apply the Boué\UDupuis
  formula and this kind of behavior will be crucial. Let
  <math|w\<in\>L<rsup|2><around*|(|<around*|[|0,\<infty\>|)>\<times\>\<Lambda\>|)>>
  and denote

  <\equation*>
    u<rsup|w>\<assign\>U<around*|(|<wide|W|~>+I<around*|(|w|)>|)><space|1em><text|and><space|2em>h<rsup|w>\<assign\>U<around*|(|<wide|W|~>+I<around*|(|w|)>|)>+w=u<rsup|w>+w.
  </equation*>

  The process <math|h<rsup|w>> satisfies\ 

  <\equation*>
    h<rsup|w>-w=u<rsup|w>=\<Xi\><around*|(|<wide|W|~>+I<around*|(|w|)>,u<rsup|w>|)>.
  </equation*>

  More explicitly, for all <math|s\<geqslant\>0> we have

  <\equation*>
    <tabular|<tformat|<table|<row|<cell|h<rsup|w><rsub|s>-w<rsub|s>>|<cell|=>|<cell|-4\<lambda\>J<rsub|s><around|\<llbracket\>|<wide|W|~><rsub|s><rsup|3>|\<rrbracket\>>-12\<lambda\>J<rsub|s><around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>I<rsub|s><around*|(|w|)>-12\<lambda\>J<rsub|s><wide|W|~><rsub|s><around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>-4\<lambda\>J<rsub|s><around*|(|I<rsub|s><around*|(|w|)>|)><rsup|3>>>|<row|<cell|>|<cell|>|<cell|-12\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|<around|\<llbracket\>|<wide|W|~><rsup|2><rsub|s>|\<rrbracket\>>\<succ\><rsub|>I<rsub|s><rsup|\<flat\>><around*|(|u<rsup|w>|)>|)>-24\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>><around*|(|J<rsub|s><around*|(|<wide|W|~><rsub|s>I<rsub|s><around*|(|w|)>\<succ\>I<rsup|\<flat\>><rsub|s><around*|(|u<rsup|w>|)>|)>|)>>>|<row|<cell|>|<cell|>|<cell|-12\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|<around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>\<succ\>I<rsup|\<flat\>><rsub|s><around*|(|u<rsup|w>|)>|)>>>|<row|<cell|>|<cell|>|<cell|-<big|sum><rsub|i=0><rsup|n><choose|n|i>J<rsub|s><around|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><wide|W|~><rsub|s>|)><rsup|i>|\<rrbracket\>><around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|s><around*|(|w|)>|)><rsup|n-i>.>>>>>
  </equation*>

  Decomposing\ 

  <\equation*>
    <around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>I<rsub|s><around*|(|w|)>=<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<succ\>\<theta\><rsub|s>I<rsub|s><around*|(|w|)>+<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<succ\><around*|(|1-\<theta\><rsub|s>|)>I<rsub|s><around*|(|w|)>+<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<circ\>I<rsub|s><around*|(|w|)>+<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<prec\>I<rsub|s><around*|(|w|)>,
  </equation*>

  we can write\ 

  <\equation>
    u<rsup|w>=U<around*|(|<wide|W|~>+I<around*|(|w|)>|)>=-4\<lambda\>J<rsub|s><around|\<llbracket\>|<wide|W|~><rsub|s><rsup|3>|\<rrbracket\>>-12\<lambda\>J<rsub|s><around|(|<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|h<rsup|w>|)>|)>+r<rsub|s><rsup|w>,<label|eq:uw>
  </equation>

  with\ 

  <\equation>
    <tabular|<tformat|<table|<row|<cell|r<rsub|s><rsup|w>>|<cell|=>|<cell|-12\<lambda\>J<rsub|s><around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<succ\><around*|(|1-\<theta\><rsub|s>|)>I<rsub|s><around*|(|w|)>-12\<lambda\>J<rsub|s><around*|(|<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<circ\>I<rsub|s><around*|(|w|)>|)>-12\<lambda\>J<rsub|s><around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<prec\>I<rsub|s><around*|(|w|)>>>|<row|<cell|>|<cell|>|<cell|-12\<lambda\>J<rsub|s><wide|W|~><rsub|s><around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>-4\<lambda\>J<rsub|s><around*|(|I<rsub|s><around*|(|w|)>|)><rsup|3>-24\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>><around*|(|J<rsub|s><around*|(|<wide|W|~><rsub|s>I<rsub|s><around*|(|w|)>\<succ\>\<theta\><rsub|s><rsub|>I<rsup|\<flat\>><rsub|s><around*|(|u<rsup|w>|)>|)>|)>>>|<row|<cell|>|<cell|>|<cell|-12\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<geqslant\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|<around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>\<succ\>I<rsup|\<flat\>><rsub|s><around*|(|u<rsup|w>|)>|)>+12\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<less\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|<around|\<llbracket\>|<wide|W|~><rsub|s><rsup|2>|\<rrbracket\>>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|u<rsup|w>|)>|)>>>|<row|<cell|>|<cell|>|<cell|-<big|sum><rsub|i=0><rsup|n><choose|n|i>J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|[|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|s>|)><rsup|i>|\<rrbracket\>><around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|s><around*|(|w|)>|)><rsup|n-i><rsub|>|]>.>>>>><label|eq:rw>
  </equation>

  The first two terms in<nbsp><eqref|eq:uw> will be used for renormalization
  while the remainder <math|r<rsup|w>> contains terms of higher regularity
  which will have to be estimated in the sequel.

  <\render-proof|Proof of Lemma<nbsp><reference|lemma:localestimate>>
    Observe that\ 

    <\equation*>
      <with|math-font|Bbb*|1><rsub|<around*|{|<around*|\<\|\|\>|W<rsub|\<infty\>>|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<leqslant\>K|}>>\<lesssim\><rsub|K,n>exp<around*|(|-<around*|\<\|\|\>|W<rsub|\<infty\>>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>|)>=exp<around*|(|-<around*|\<\|\|\>|<wide|W|~><rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|U<around*|(|<wide|W|~>|)>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>|)>
    </equation*>

    and\ 

    <\equation*>
      <around*|\||D<rsub|T>|\|><rsup|p>=e<rsup|-p
      <around*|[|V<rsub|T><around*|(|<wide|W|~><rsub|T>+I<around*|(|U<around*|(|<wide|W|~>|)>|)>|)>+<big|int><rsup|T><rsub|0>U<around*|(|<wide|W|~>|)>\<mathd\><wide|X|~>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|U<rsub|t><around*|(|<wide|W|~>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>.
    </equation*>

    Combining these two facts we have

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|>>|<row|<cell|\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|\||D<rsub|T>|\|><rsup|p><with|math-font|Bbb*|1><rsub|<around*|{|<around*|\<\|\|\>|W|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<leqslant\>K|}>>|]>>>|<row|<cell|\<lesssim\>\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|exp<around*|(|-p
      <around*|(|V<rsub|T><around*|(|<wide|W|~><rsub|T>+I<rsub|T><around*|(|U<around*|(|<wide|W|~>|)>|)>|)>+<big|int><rsup|T><rsub|0>U<rsub|t><around*|(|<wide|W|~>|)>\<mathd\><wide|X<rsub|t><rsub|>|~>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|U<rsub|t><around*|(|<wide|W|~>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|)>|\<nobracket\>>|\<nobracket\>>>>|<row|<cell|<space|2em><space|2em><space|2em><space|2em><space|2em><around*|<left|.>|<around*|<left|.>|-<around*|\<\|\|\>|<wide|W|~><rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|U<around*|(|<wide|W|~>|)>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>|<right|)|3>>|<right|]|3>>>>|<row|<cell|=\<bbb-E\><around*|[|exp<around*|(|-p
      <around*|(|V<rsub|T><around*|(|W<rsub|T>+I<rsub|T><around*|(|U<around*|(|W|)>|)>|)>+<big|int><rsup|T><rsub|0>U<rsub|t><around*|(|W|)>\<mathd\>X<rsub|t>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|U<rsub|t><around*|(|W|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|)>|\<nobracket\>>|\<nobracket\>>>>|<row|<cell|<space|2em><space|2em><space|2em><space|2em><space|2em><around*|<left|.>|<around*|<left|.>|-<around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|U<around*|(|W|)>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>|<right|)|3>>|<right|]|3>>.>>>>>
    </equation*>

    The Boué\UDupuis formula<nbsp><eqref|eq:bd> provides the variational
    bound

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|-log
      \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|\||D<rsub|T>|\|><rsup|p><with|math-font|Bbb*|1><rsub|<around*|{|<around*|\<\|\|\>|W|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<leqslant\>K|}>>|]>>>|<row|<cell|<space|2em>\<gtrsim\>inf<rsub|w\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|p
      <around*|(|V<rsub|T><around*|(|W<rsub|T>+I<rsub|T><around*|(|h<rsup|w>|)>|)>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|h<rsup|w>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|)>|\<nobracket\>>>>|<row|<cell|<space|2em>\<gtrsim\><space|2em><space|2em><around*|\<nobracket\>|+<frac|1-p|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t+<around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<frac|1|2><big|int><rsup|\<infty\>><rsub|T><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>>>>
    </equation*>

    where we have set <math|h<rsup|w>=w+U<around*|(|W+I<around*|(|w|)>|)>> as
    above. Recall now that from Theorem<nbsp><reference|thm:equicoerc> there
    exists a constant <math|C>, independent of <math|T> , such that for each
    <math|h<rsup|w>>,

    <\equation*>
      \<bbb-E\><around*|[|p <around*|(|V<rsub|T><around*|(|W<rsub|T>+I<rsub|T><around*|(|h<rsup|w>|)>|)>+<frac|1|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|h<rsup|w>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|)>|]>\<geqslant\>-C+<frac|1|4>\<bbb-E\><rsub|\<bbb-P\>><around*|[|\<lambda\><around*|\<\|\|\>|I<rsub|T><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<big|int><rsup|T><rsub|0><around*|\<\|\|\>|l<rsup|T><around*|(|h<rsup|w>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>|]>
    </equation*>

    where

    <\equation*>
      l<rsub|t><rsup|T><around*|(|h<rsup|w>|)>=h<rsub|t><rsup|w>+\<lambda\><with|math-font|Bbb*|1><rsub|t\<leqslant\>T>\<bbb-W\><rsup|<around*|\<langle\>|3|\<rangle\>>><rsub|t>+\<lambda\><with|math-font|Bbb*|1><rsub|t\<leqslant\>T>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|2>\<succ\>I<rsub|t><rsup|\<flat\>><around*|(|h<rsup|w>|)>|)>.
    </equation*>

    Using eq.<nbsp><eqref|eq:uw> we compute

    <\eqnarray*>
      <tformat|<table|<row|<cell|<with|math-font|Bbb*|1><rsub|t\<leqslant\>T>*l<rsub|t><rsup|T><around*|(|h<rsup|w>|)>>|<cell|=>|<cell|<with|math-font|Bbb*|1><rsub|t\<leqslant\>T>*h<rsub|t><rsup|w>+\<lambda\><with|math-font|Bbb*|1><rsub|t\<leqslant\>T>\<bbb-W\><rsup|<around*|\<langle\>|3|\<rangle\>>><rsub|t>+\<lambda\><with|math-font|Bbb*|1><rsub|t\<leqslant\>T>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|2>\<succ\>I<rsub|t><rsup|\<flat\>><around*|(|h<rsup|w>|)>|)>>>|<row|<cell|>|<cell|=>|<cell|<with|math-font|Bbb*|1><rsub|t\<leqslant\>T><around*|(|u<rsub|t><rsup|w>+w<rsub|t>|)>+\<lambda\><with|math-font|Bbb*|1><rsub|t\<leqslant\>T>\<bbb-W\><rsup|<around*|\<langle\>|3|\<rangle\>>><rsub|t>+\<lambda\><with|math-font|Bbb*|1><rsub|t\<leqslant\>T>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|2>\<succ\>I<rsub|t><rsup|\<flat\>><around*|(|h<rsup|w>|)>|)>>>|<row|<cell|>|<cell|=>|<cell|<with|math-font|Bbb*|1><rsub|t\<leqslant\>T><around*|(|r<rsub|t><rsup|w>+w<rsub|t>|)>.>>>>
    </eqnarray*>

    At this point we need a lower bound for

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|\<bbb-E\><around*|[|<frac|1|4><around*|(|\<lambda\><around*|\<\|\|\>|I<rsub|T><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<big|int><rsup|T><rsub|0><around*|\<\|\|\>|r<rsub|t><rsup|w>+w<rsub|t>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>t|)>+<frac|1-p|2><big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|\<nobracket\>>>>|<row|<cell|<space|2em><space|2em><space|2em><space|2em><space|2em><around*|\<nobracket\>|+<around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<frac|1|2><big|int><rsup|\<infty\>><rsub|T><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>-C.>>>>>
    </equation*>

    Given that we need to take <math|p\<gtr\>1>, this expression present a
    difficulty in the fact that the term <math|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t>
    appears with a negative coefficient. Note that this term cannot easily be
    controlled via <math|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|r<rsub|t><rsup|w>+w<rsub|t>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>t>
    since the contribution <math|r<rsup|w>>, see eq.<nbsp><eqref|eq:rw>,
    contains factors which are homogeneous in <math|w> of order up to
    <math|3>. This is the reason we had to localize the estimate, introduce
    the \Pgood\Q term <math|<around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>>,
    and introduce the term <math|J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|(|<around*|\<llbracket\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|s>|\<rrbracket\>><rsup|n>|)>>
    in <eqref|eq:pre-pre-drift> which will help us to control the growth of
    <math|r<rsup|w>>. Indeed in Lemma<nbsp><reference|lemma:driftgronwallestimate>
    below, a Gronwall argument will allow us to show that
    <math|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t>
    can be bounded by a combination of the other \Pgood\Q terms as

    <\equation*>
      \<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>\<lesssim\>\<bbb-E\><around*|[|<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|T><around*|(|h|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<around*|\<\|\|\>|I<rsub|T><rsup|\<flat\>><rsup|><around*|(|h|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|t>+r<rsub|t><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>t+1|]>.
    </equation*>

    This implies that for <math|1\<less\>p\<ll\>2>,

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|-log\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|\||D<rsub|T>|\|><rsup|p><with|math-font|Bbb*|1><rsub|<around*|{|<around*|\<\|\|\>|W|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<leqslant\>K|}>>|]>>>|<row|<cell|<space|1em>\<geqslant\>inf<rsub|w\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|{|<frac|1|4><around*|[|\<lambda\><around*|\<\|\|\>|I<rsub|T><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<big|int><rsup|T><rsub|0><around*|\<\|\|\>|l<rsup|T><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>t|]>|\<nobracket\>>>>|<row|<cell|<space|1em><space|1em><space|1em>+<around*|(|1-p|)>C<around*|[|<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|T><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<around*|\<\|\|\>|I<rsub|T><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<big|int><rsup|T><rsub|0><around*|\<\|\|\>|l<rsup|T><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>t|]>>>|<row|<cell|<space|1em><space|1em><space|1em><around*|<left|.>|+<around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>|<right|}|3>>-C>>|<row|<cell|<space|1em>\<geqslant\>-C>>>>>
    </equation*>

    which gives the claim. Note that here we used the bound

    <\equation*>
      \<bbb-E\><around*|\<\|\|\>|I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<lesssim\>\<bbb-E\><around*|\<\|\|\>|W<rsub|\<infty\>>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+\<bbb-E\><around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>
    </equation*>

    <\equation*>
      \<lesssim\>C+\<bbb-E\><around*|\<\|\|\>|W<rsub|\<infty\>>+I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>
    </equation*>

    as well as the fact that <math|<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>\<lesssim\><around*|\<\|\|\>|I<rsub|\<infty\>><around*|(|h<rsup|w>|)>|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>>
    to conclude.
  </render-proof>

  The following lemmas complete the proof.

  <\lemma>
    <label|lemma:driftgronwallestimate>For <math|n\<in\>\<bbb-N\>> odd and
    large enough\ 

    <\equation*>
      \<bbb-E\><big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s\<lesssim\>\<bbb-E\><big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>+r<rsub|s><rsup|w>|\<\|\|\>><rsup|2>\<mathd\>s+\<bbb-E\><around*|\<\|\|\>|I<rsub|T><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|T><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+1.
    </equation*>
  </lemma>

  <\proof>
    Let us introduce the notation\ 

    <\equation*>
      Aux<rsub|s><around*|(|W,w|)>\<assign\><big|sum><rsub|i=0><rsup|n><choose|n|i>J<rsub|s><around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2><around*|(|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|s>|)><rsup|i>|\<rrbracket\>><around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|s><around*|(|w|)>|)><rsup|n-i><rsub|>|)>.
    </equation*>

    Write <math|r<rsub|s><rsup|w>=<wide|r|~><rsub|s><rsup|w>+Aux<around*|(|W,w|)>>
    and observe that

    <\eqnarray*>
      <tformat|<table|<row|<cell|w<rsup|2><rsub|s>>|<cell|=>|<cell|2<around*|(|w<rsub|s>+r<rsub|s><rsup|w>|)><rsup|2>-4w<rsub|s>r<rsub|s><rsup|w>-2<around*|(|r<rsub|s><rsup|w>|)><rsup|2>-w<rsup|2><rsub|s>>>|<row|<cell|>|<cell|=>|<cell|2<around*|(|w<rsub|s>+r<rsub|s><rsup|w>|)><rsup|2>-4w<rsub|s><wide|r|~><rsub|s><rsup|w>-2<around*|(|r<rsub|s><rsup|w>|)><rsup|2>-w<rsup|2><rsub|s>-4Aux<around*|(|W,w|)>w<rsub|s>.>>>>
    </eqnarray*>

    We can apply Ito formula to obtain\ 

    <\equation*>
      <big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>Aux<rsub|s><around*|(|W,w|)>w<rsub|s>ds=<wide|Aux|\<bar\>><rsub|T><around*|(|W,w|)>+<text|martingale>
    </equation*>

    where

    <\equation*>
      <wide|Aux|\<bar\>><rsub|T><around*|(|W,w|)>\<assign\><big|sum><rsub|i=0><rsup|n><frac|1|n+1-i><choose|n|i><big|int><rsub|\<Lambda\>><around*|(|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|T>|)><rsup|i>|\<rrbracket\>><around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|T><around*|(|w|)>|)><rsup|n+1-i><rsub|>|)>.
    </equation*>

    By Lemma<nbsp><reference|lemma:bounds2> below, we have constants
    <math|c,C> and a random variable <math|Q<rsub|T><around*|(|W|)>> such
    that

    <\equation*>
      sup<rsub|T>\<bbb-E\><around*|[|<around*|\||Q<rsub|T><around*|(|W|)>|\|>|]>\<less\>\<infty\>
    </equation*>

    and

    \;

    <\equation*>
      c<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+c<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>-Q<rsub|T><around*|(|W|)>\<leqslant\><big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|T><around*|(|W,w|)>
    </equation*>

    <\equation*>
      \<leqslant\>C<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>+C<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+Q<rsub|T><around*|(|W|)>.
    </equation*>

    Integrating over space we obtain\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|<frac|\<mathd\>|\<mathd\>t>\<bbb-E\><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|t><around*|(|W,w|)>|)>>|<cell|\<leqslant\>>|<cell|2*\<bbb-E\><around*|\<\|\|\>|w<rsub|t><rsub|>+r<rsub|t><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>-4*\<bbb-E\><big|int>w<rsub|s><wide|r|~><rsub|s><rsup|w>-<around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|2*\<bbb-E\><around*|\<\|\|\>|w<rsub|t><rsub|>+r<rsub|t><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>+4*\<bbb-E\><around*|\<\|\|\>|<wide|r|~><rsub|s><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>.>>>>
    </eqnarray*>

    Now by Lemma<nbsp><reference|lemma:estimateremainder> below

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>><around*|\<\|\|\>|<wide|r|~><rsub|t><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>>|<cell|\<lesssim\>>|<cell|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>+<around*|\<\|\|\>|I<rsub|t><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<rsub|>Q<rsub|t><around*|(|W|)>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|Q<rsub|t><around*|(|W|)>+<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|t><around*|(|W,w|)>+<around*|\<\|\|\>|I<rsub|t><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>>>>>>
    </equation*>

    Gathering things together we have

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|>|<cell|<frac|\<mathd\>|\<mathd\>t>\<bbb-E\><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|t><around*|(|W,w|)>|)>>>|<row|<cell|\<lesssim\>>|<cell|<frac|1|<around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>>><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|t><around*|(|W,w|)>|)>+<frac|1|<around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>>><around*|(|<around*|\<\|\|\>|I<rsub|t><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)>>>|<row|<cell|>|<cell|+2*\<bbb-E\><around*|\<\|\|\>|w<rsub|t><rsub|>+r<rsub|t><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>.>>>>>
    </equation*>

    Gronwall's lemma allows then to conclude\ 

    <\equation*>
      -1+\<bbb-E\><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s|)>\<lesssim\>\<bbb-E\><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|t><around*|(|W,w|)>|)>
    </equation*>

    <\equation*>
      \<lesssim\>\<bbb-E\><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s><rsub|>+r<rsub|s><rsup|w>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)>+1.
    </equation*>

    \;
  </proof>

  <\lemma>
    <label|lemma:bounds2>There exists constants <math|c,C> and a random
    variable <math|Q<rsub|T><around*|(|W|)>> such that

    <\equation*>
      sup<rsub|T>\<bbb-E\><around*|[|<around*|\||Q<rsub|T><around*|(|W|)>|\|>|]>\<less\>\<infty\>,
    </equation*>

    and

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|-Q<rsub|T><around*|(|W|)>+c<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+c<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<wide|Aux|\<bar\>><rsub|T><around*|(|W,w|)>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|C<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>+C<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+Q<rsub|T><around*|(|W|)>>>>>
    </eqnarray*>

    \;
  </lemma>

  <\proof>
    We recall that\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|<wide|Aux|\<bar\>><rsub|T><around*|(|W,w|)>>|<cell|=>|<cell|<big|sum><rsub|i=0><rsup|n><frac|1|n+1-i><choose|n|i><big|int><around*|(|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|T>|)><rsup|i>|\<rrbracket\>><rsup|><around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|T><around*|(|w|)>|)><rsup|n+1-i><rsub|>|)>>>|<row|<cell|>|<cell|=>|<cell|<big|sum><rsub|i=1><rsup|n><frac|1|n+1-i><choose|n|i><big|int><around*|(|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|T>|)><rsup|i>|\<rrbracket\>><rsup|><around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|T><around*|(|w|)>|)><rsup|n+1-i><rsub|>|)>>>|<row|<cell|>|<cell|>|<cell|+<frac|1|n+1><around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>>>>>
    </eqnarray*>

    and since <math|sup<rsub|T\<less\>\<infty\>>\<bbb-E\><around*|[|<around*|\<\|\|\>|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>W<rsub|T>|)><rsup|i>|\<rrbracket\>>|\<\|\|\>><rsup|p><rsub|<value|VV><rsup|-\<varepsilon\>>>|]>\<less\>\<infty\>>
    for any <math|p\<less\>\<infty\>> and any <math|\<varepsilon\>\<gtr\>0>
    it is enough to bound <math|<around*|\<\|\|\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|T><around*|(|w|)>|)><rsup|n+1-i><rsub|>|\<\|\|\>><rsup|q><rsub|B<rsub|1,1><rsup|\<varepsilon\>>>>
    for some <math|q\<gtr\>1> by the terms
    <math|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>>
    and <math|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>\<lesssim\><big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s>.
    By interpolation we can estimate, for <math|i\<geqslant\>1>,

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|<around*|(|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|T><around*|(|w|)>|)><rsup|n+1-i><rsub|>|\<\|\|\>><rsub|B<rsub|1,1><rsup|\<varepsilon\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|n><rsub|B<rsup|\<varepsilon\>><rsub|n,1>>+C>>|<row|<cell|<space|2em>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsub|W<rsup|-1/2,n+1>><rsup|n-<frac|1|<around*|(|n-1|)>>><rsub|><around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|<frac|1|n-1>><rsub|H<rsup|1>>+C<space|2em><space|2em><text|(let
      <math|\<varepsilon\>=<tfrac|1|n<around*|(|n-1|)>>>)>>>>>>
    </equation*>

    Choosing <math|q=n/<around*|(|n-<frac|1|<around*|(|n-1|)>>|)>\<gtr\>1>,
    we have\ 

    <\equation*>
      <around*|(|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsub|W<rsup|-1/2,n+1>><rsup|n-<frac|1|<around*|(|n-1|)>>><rsub|><around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|<frac|1|n-1>><rsub|H<rsup|1>>|)><rsup|q>=<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsub|W<rsup|-1/2,n+1>><rsup|n><rsub|><around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|<frac|n|<around*|(|n-1|)>n-1>><rsub|H<rsup|1>>.
    </equation*>

    Now for <math|n> large enough <math|<frac|n|<around*|(|n-1|)>n-1>\<leqslant\><frac|2|n+1>>
    and using Young's inequality we can estimate

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsub|W<rsup|-1/2,n+1>><rsup|n><rsub|><around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|<frac|n|<around*|(|n-1|)>n-1>><rsub|H<rsup|1>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsub|W<rsup|-1/2,n+1>><rsup|n><rsub|><around*|(|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|<frac|2|n+1>><rsub|H<rsup|1>>+1|)>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsub|W<rsup|-1/2,n+1>><rsup|n+1><rsub|>+<around*|\<\|\|\>|I<rsub|T><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+1>>>>>
    </equation*>

    \;
  </proof>

  <\lemma>
    <label|lemma:estimateremainder>Let

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<wide|r|~><rsub|s><rsup|w>>|<cell|=>|<cell|-12\<lambda\>J<rsub|s><around*|\<llbracket\>|W<rsub|s><rsup|2>|\<rrbracket\>>\<succ\><around*|(|1-\<theta\><rsub|s>|)>I<rsub|s><around*|(|w|)>+12\<lambda\>J<rsub|s><around*|(|<around*|\<llbracket\>|W<rsub|s><rsup|2>|\<rrbracket\>>\<circ\>I<rsub|s><around*|(|w|)>|)>+12\<lambda\>J<rsub|s><around*|\<llbracket\>|W<rsub|s><rsup|2>|\<rrbracket\>>\<prec\>I<rsub|s><around*|(|w|)>>>|<row|<cell|>|<cell|>|<cell|-12\<lambda\>J<rsub|s>W<rsub|s><around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>-4\<lambda\>J<rsub|s><around*|(|I<rsub|s><around*|(|w|)>|)><rsup|3>-24\<lambda\><around*|(|J<rsub|s><around*|(|W<rsub|s>I<rsub|s><around*|(|w|)>\<succ\>\<theta\><rsub|s><rsub|>I<rsup|\<flat\>><rsub|s><around*|(|u<rsup|w>|)>|)>|)>>>|<row|<cell|>|<cell|>|<cell|-12\<lambda\>J<rsub|s><around*|(|<around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>\<succ\>\<theta\><rsub|s>I<rsup|\<flat\>><rsub|s><around*|(|u<rsup|w>|)>|)>+\<lambda\><with|math-font|Bbb*|1><rsub|<around*|{|s\<less\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|\<bbb-W\><rsub|s><rsup|2>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|u<rsup|w>|)>|)>.>>>>>
    </equation*>

    Setting <math|h<rsup|w>=u+w>, there exists a random variable
    <math|Q<rsub|t><around*|(|W|)>> such that
    <math|sup<rsub|t>\<bbb-E\><around*|[|<around*|\||Q<rsub|t><around*|(|W|)>|\|>|]>\<less\>\<infty\>>
    and

    <\equation*>
      <around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>><around*|\<\|\|\>|<wide|r|~><rsub|t><rsup|w>|\<\|\|\>><rsup|2>\<lesssim\><big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>+<around|\<\|\|\>|I<rsub|t><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<rsub|>Q<rsub|t><around*|(|W|)>.
    </equation*>
  </lemma>

  <\proof>
    Note that\ 

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|<with|math-font|Bbb*|1><rsub|<around*|{|s\<less\><wide|T|\<bar\>>|}>>J<rsub|s><around*|(|\<bbb-W\><rsub|s><rsup|2>\<succ\>I<rsub|s><rsup|\<flat\>><around*|(|u<rsup|w>|)>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>>|<cell|\<lesssim\><rsub|<wide|T|\<bar\>>>>|<cell|<frac|1|<around*|\<langle\>|s|\<rangle\>><rsup|2>><around*|\<\|\|\>|\<bbb-W\><rsub|s><rsup|2>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><around*|\<\|\|\>|I<rsub|s><rsup|\<flat\>><around*|(|u<rsup|w>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|4>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<frac|1|<around*|\<langle\>|s|\<rangle\>><rsup|2>><around*|(|<around*|\<\|\|\>|\<bbb-W\><rsub|s><rsup|2>|\<\|\|\>><rsup|4><rsub|<value|VV><rsup|-1-\<varepsilon\>>>+<around*|\<\|\|\>|I<rsub|s><rsup|\<flat\>><around*|(|u<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)>.>>>>>
    </equation*>

    Moreover <math|h<rsup|w>=u<rsup|w>+w> implies

    <\equation*>
      <around|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>\<lesssim\><around|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>+<around|\<\|\|\>|I<rsub|t><rsup|\<flat\>><rsup|><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|n+1><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>,
    </equation*>

    and <math|<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>\<lesssim\><rsub|><around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|h<rsup|w>|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>>
    . From Lemma<nbsp><reference|L4estimate> we get

    <\equation*>
      <around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>\<lesssim\>C+<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>.
    </equation*>

    The estimation for the other terms is easy but technical and postponed
    until Section<nbsp><reference|sec:analytic>.
  </proof>

  <section|Singularity of <math|\<Phi\><rsub|3><rsup|4>> w.r.t. the free
  field>

  The goal of this section is to prove that the
  <math|\<Phi\><rsub|3><rsup|4>> measure is singular with respect to the
  Gaussian free field. For this we have to find a set
  <math|S\<subseteq\><value|VV><rsup|-1/2-\<varepsilon\>><around*|(|\<Lambda\>|)>>
  such that <math|\<bbb-P\><around*|(|W<rsub|\<infty\>>\<in\>S|)>=1> and
  <math|\<bbb-Q\><rsup|u><around*|(|W<rsub|\<infty\>>\<in\>S|)>=0>. Together
  with<nbsp><eqref|eq:Phi4def>, this will imply singularity. We claim that
  setting\ 

  <\equation*>
    S\<assign\><around*|{|f\<in\><value|VV><rsup|-1/2-\<varepsilon\>><around*|(|\<Lambda\>|)>:<frac|1|T<rsub|n><rsup|1/2+\<delta\>>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T<rsub|n>>f|)><rsup|4>|\<rrbracket\>>\<rightarrow\>0|}>
  </equation*>

  for some suitable subsequence <math|T<rsub|n>>, does the job. Here

  <\equation*>
    <around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>f|)><rsup|4>|\<rrbracket\>>=<around*|(|\<theta\><rsub|T>f|)><rsup|4>-6\<bbb-E\><around*|[|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>><around*|(|0|)>|)><rsup|2>|]><around*|(|\<theta\><rsub|T>f|)><rsup|2>+3\<bbb-E\><around*|[|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>><around*|(|0|)>|)><rsup|2>|]><rsup|2>
  </equation*>

  denotes the Wick ordering with respect to the Gaussian free field. Let us
  prove first that indeed <math|\<bbb-P\><around*|(|W<rsub|\<infty\>>\<in\>S|)>=1>
  for some <math|T<rsub|n>>. For later use we note that\ 

  <\equation*>
    \<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>=4<around*|(|\<theta\><rsub|T>W<rsub|t>|)><rsup|3>-12\<bbb-E\><around*|[|<around*|(|\<theta\><rsub|T>W<rsub|t><around*|(|0|)>|)><rsup|2>|]><around*|(|\<theta\><rsub|T>W<rsub|t>|)>
  </equation*>

  and

  <\equation*>
    \<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,2>=12<around*|(|<around*|(|\<theta\><rsub|T>W<rsub|t>|)><rsup|2>-\<bbb-E\><around*|[|<around*|(|\<theta\><rsub|T>W<rsub|t><around*|(|0|)>|)><rsup|2>|]>|)>.
  </equation*>

  <\lemma>
    For any <math|\<delta\>\<gtr\>0>

    <\equation*>
      lim<rsub|T\<rightarrow\>\<infty\>>\<bbb-E\><around*|[|<around*|(|<frac|1|T<rsup|<around*|(|1+\<delta\>|)>/2>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>|)><rsup|2>|]>=0.
    </equation*>
  </lemma>

  <\proof>
    Wick products corresponds to iterated Ito integrals. Introducing the
    notation

    <\equation*>
      \<mathd\>w<rsub|t><rsup|\<theta\><rsub|T>>=\<theta\><rsub|T>J<rsub|t>\<mathd\>X<rsub|t>,
    </equation*>

    we can verify by Ito formula that

    <\equation*>
      <big|int><rsub|\<Lambda\>><around*|\<llbracket\>|\<theta\><rsub|T>W<rsub|\<infty\>><rsup|4>|\<rrbracket\>>=<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>w<rsup|\<theta\><rsub|T>><rsub|t><rsub|>=<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsub|t>.
    </equation*>

    Since <math|\<theta\><rsub|T>J<rsub|t>=0> for <math|t\<geqslant\>T>, Ito
    isometry gives

    <\equation*>
      \<bbb-E\><around*|\||<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsub|t>|\|><rsup|2>=\<bbb-E\><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)><rsup|2>\<mathd\>t.
    </equation*>

    Then, again by Ito formula the expectation on the r.h.s. can be estimated
    as

    <\eqnarray*>
      <tformat|<table|<row|<cell|\<bbb-E\><around*|[|<big|int><rsub|\<Lambda\>><around*|(|\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)><rsup|2>|]>>|<cell|=>|<cell|4\<bbb-E\><around*|[|<around*|\||<big|sum><rsub|k<rsub|1>,k<rsub|2>,k<rsub|3>><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0>\<mathd\>w<rsub|s<rsub|1>><rsup|\<theta\><rsub|T>><around*|(|k<rsub|1>|)>\<mathd\>w<rsub|s<rsub|2>><rsup|\<theta\><rsub|T>><around*|(|k<rsub|2>|)>\<mathd\>w<rsup|\<theta\><rsub|T>><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|\|><rsup|2>|]>>>|<row|<cell|>|<cell|=>|<cell|24\<bbb-E\><around*|[|<big|sum><rsub|k<rsub|1>,k<rsub|2>,k<rsub|3>><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0><frac|\<theta\><rsup|2><rsub|T><around*|(|k<rsub|1>|)>\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)><rsub|>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<theta\><rsub|T><rsup|2><around*|(|k<rsub|2>|)>\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<theta\><rsup|2><rsub|T><around*|(|k<rsub|3>|)>\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>|]>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|24\<bbb-E\><around*|[|<big|sum><rsub|k<rsub|1>,k<rsub|2>,k<rsub|3>><big|int><rsup|t><rsub|0><big|int><rsup|t><rsub|0><big|int><rsup|t><rsub|0><frac|\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)><rsub|>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>|]>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|t<rsup|3>>>>>
    </eqnarray*>

    Now recall that <math|<around*|\<\|\|\>|J<rsub|t>f|\<\|\|\>><rsub|L<rsup|2><around*|(|\<Lambda\>|)>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|\<\|\|\>|f|\<\|\|\>><rsub|L<rsup|2><around*|(|\<Lambda\>|)>>>
    to conclude:

    <\equation*>
      \<bbb-E\><around*|[|<frac|1|T<rsup|1+\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)><rsup|2>\<mathd\>t|]>\<leqslant\><frac|1|T<rsup|1+\<delta\>>><big|int><rsup|T><rsub|0><frac|1|t<rsup|3>>\<bbb-E\><around*|[|<around*|\<\|\|\>|<around*|(|\<theta\><rsub|T>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>|\<\|\|\>><rsub|L<rsup|2><around*|(|\<Lambda\>|)>><rsup|2>|]>\<mathd\>t\<rightarrow\>0
    </equation*>
  </proof>

  The lemma implies that <math|<frac|1|T<rsup|<around*|(|1+\<delta\>|)>/2>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>\<rightarrow\>0>
  in <math|L<rsup|2><around*|(|\<bbb-P\>|)>>. So there exists a subsequence
  <math|T<rsub|n>> such that <math|<frac|1|T<rsub|n><rsup|<around*|(|1+\<delta\>|)>/2>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T<rsub|n>>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>\<rightarrow\>0>
  almost surely.

  \;

  The next step of the proof is to check that
  <math|\<bbb-Q\><rsup|u><around*|(|W<rsub|\<infty\>>\<in\>S|)>=0>. More
  concretely we will show that for a subsequence of <math|T<rsub|n>> (not
  relabeled)

  <\equation*>
    <frac|1|T<rsub|n><rsup|1-\<delta\>>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T<rsub|n>>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>\<rightarrow\>-\<infty\>,
  </equation*>

  <math|\<bbb-Q\><rsup|u>> almost surely. Observe that\ 

  <\equation*>
    <tabular|<tformat|<table|<row|<cell|<big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>>|<cell|=>|<cell|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsub|t>>>|<row|<cell|>|<cell|=>|<cell|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsup|u><rsub|t>+<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>u<rsub|t>\<mathd\>t>>|<row|<cell|>|<cell|=>|<cell|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsup|u><rsub|t>-\<lambda\><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t>>|<row|<cell|>|<cell|>|<cell|-\<lambda\><big|int><rsup|\<infty\>><rsub|<wide|T|\<bar\>>><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|u,2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>\<mathd\>t>>|<row|<cell|>|<cell|>|<cell|-<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t><around*|\<langle\>|D|\<rangle\>><rsup|-1/2><around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>\<mathd\>t.>>>>>
  </equation*>

  <\equation*>
    \;
  </equation*>

  We expect the term\ 

  <\equation*>
    <big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t
  </equation*>

  to go to infinity faster than <math|T<rsup|1-\<delta\>>>,
  <math|\<bbb-Q\><rsup|u>>-almost surely. To actually prove it, we start by a
  computation in average.

  <\lemma>
    It holds

    <\equation*>
      lim<rsub|T\<rightarrow\>\<infty\>><frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t|]>=\<infty\>.
    </equation*>
  </lemma>

  <\proof>
    Recall that <math|\<mathd\>w<rsub|t><rsup|\<theta\><rsub|T>>=\<theta\><rsub|T>J<rsub|t>\<mathd\>X<rsub|t>>.
    With a slight abuse of notation we can write

    <\math>
      <\equation*>
        <tabular|<tformat|<table|<row|<cell|>|<cell|>|<cell|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t>>|<row|<cell|>|<cell|=>|<cell|16<big|int><rsup|\<infty\>><rsub|0><big|sum><rsub|k><frac|\<theta\><rsub|T><around*|(|k|)>\<sigma\><rsup|2><rsub|t><around*|(|k|)>|<around*|\<langle\>|k|\<rangle\>><rsup|2>><around*|(|<big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0>\<mathd\>w<rsub|s<rsub|1>><rsup|\<theta\><rsub|T>><around*|(|k<rsub|1>|)>\<mathd\>w<rsub|s<rsub|2>><rsup|\<theta\><rsub|T>><around*|(|k<rsub|2>|)>\<mathd\>w<rsup|\<theta\><rsub|T>><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|\<nobracket\>>>>|<row|<cell|>|<cell|>|<cell|\<times\><around*|\<nobracket\>|<big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0>\<mathd\>w<rsub|s<rsub|1>><around*|(|k<rsub|1>|)>\<mathd\>w<rsub|s<rsub|2>><around*|(|k<rsub|2>|)>\<mathd\>w<rsub|s<rsub|3>><around*|(|k<rsub|3>|)>
        |)>\<mathd\>t>>>>>\ 
      </equation*>
    </math>

    and by Ito isometry

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|>|<cell|>|<cell|\<bbb-E\><around|<left|[|4>|<big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0>\<mathd\>w<rsub|s<rsub|1>><rsup|\<theta\><rsub|T>><around*|(|k<rsub|1>|)>\<mathd\>w<rsub|s<rsub|2>><rsup|\<theta\><rsub|T>><around*|(|k<rsub|2>|)>\<mathd\>w<rsup|\<theta\><rsub|T>><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>
      |<right|.>>>>|<row|<cell|>|<cell|>|<cell|\<times\><big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><around*|<left|.>|<big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0>\<mathd\>w<rsub|s<rsub|1>><around*|(|k<rsub|1>|)>\<mathd\>w<rsub|s<rsub|2>><around*|(|k<rsub|2>|)>\<mathd\>w<rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<right|]|1>>>>|<row|<cell|>|<cell|=>|<cell|6<big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0><frac|\<theta\><rsub|T><around*|(|k<rsub|1>|)>\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<theta\><rsub|T><around*|(|k<rsub|2>|)>\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<theta\><rsub|T><around*|(|k<rsub|3>|)>\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>>>>>>
    </equation*>

    For <math|T> large enough and since <math|\<sigma\><rsup|2>> and
    <math|\<theta\>> are positive, we have\ 

    <\eqnarray*>
      <tformat|<cwith|1|-1|2|-1>|<table|<row|<cell|>|<cell|>|<cell|<big|int><rsup|\<infty\>><rsub|0><big|sum><rsub|k><frac|\<theta\><rsub|T><around*|(|k|)>\<sigma\><rsup|2><rsub|t><around*|(|k|)>|<around*|\<langle\>|k|\<rangle\>><rsup|2>><big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><big|int><rsup|t><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0><frac|\<theta\><rsub|T><around*|(|k<rsub|1>|)>\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<theta\><rsub|T><around*|(|k<rsub|2>|)>\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<theta\><rsub|T><around*|(|k<rsub|3>|)>\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>\<mathd\>t>>|<row|<cell|>|<cell|\<geqslant\>>|<cell|<big|int><rsup|T/2><rsub|T/8><big|sum><rsub|k><frac|\<sigma\><rsup|2><rsub|t><around*|(|k|)>|<around*|\<langle\>|k|\<rangle\>><rsup|2>><big|sum><rsub|k<rsub|1>+k<rsub|2>+k<rsub|3>=k><big|int><rsup|T/8><rsub|0><big|int><rsup|s<rsub|1>><rsub|0><big|int><rsup|s<rsub|2>><rsub|0><frac|\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>\<mathd\>t>>>>
    </eqnarray*>

    Introduce the notation <math|\<bbb-Z\><rsub|+><rsup|3>=<around*|{|n\<in\>\<bbb-Z\><rsup|3>:n=<around*|(|n<rsub|1>,n<rsub|2>,n<rsub|3>|)>
    with n<rsub|i>\<geqslant\>0|}>>. After restricting the sum to
    <math|<around*|(|\<bbb-Z\><rsub|+><rsup|3>|)><rsup|3>> we get the bound

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|\<geqslant\>>|<cell|<big|int><rsup|T/2><rsub|T/8><big|sum><rsub|k><frac|\<sigma\><rsup|2><rsub|t><around*|(|k|)>|<around*|\<langle\>|k|\<rangle\>><rsup|2>><big|sum><rsub|<tabular|<tformat|<table|<row|<cell|k<rsub|1>,k<rsub|2>,k<rsub|3>\<in\>\<bbb-Z\><rsub|+><rsup|3>>>|<row|<cell|k<rsub|1>+k<rsub|2>+k<rsub|3>=k>>>>>><big|int><rsup|T/8><rsub|3T/32><big|int><rsup|s<rsub|1>><rsub|3T/32><big|int><rsup|s<rsub|2>><rsub|3T/32><frac|\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>\<mathd\>t>>|<row|<cell|\<gtrsim\>>|<cell|<frac|1|T<rsup|2>><big|sum><rsub|k\<in\>\<bbb-Z\><rsub|+><rsup|3>><around*|(|\<rho\><rsub|T/2><around*|(|k|)>-\<rho\><rsub|T/8><around*|(|k|)>|)><big|sum><rsub|<tabular|<tformat|<table|<row|<cell|k<rsub|1>,k<rsub|2>,k<rsub|3>\<in\>\<bbb-Z\><rsub|+><rsup|3>>>|<row|<cell|k<rsub|1>+k<rsub|2>+k<rsub|3>=k>>>>>><big|int><rsup|T/8><rsub|3T/32><big|int><rsup|s<rsub|1>><rsub|3T/32><big|int><rsup|s<rsub|2>><rsub|3T/32><frac|\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>>>>>>
    </equation*>

    Now, for large enough <math|T> if <math|k<rsub|1>+k<rsub|2>+k<rsub|3>=k>
    and <math|<around*|\<langle\>|k<rsub|i>|\<rangle\>>\<leqslant\>T/8> then
    <math|<around*|\<langle\>|k|\<rangle\>>\<leqslant\>T/2\<times\>0.9>.
    \ Furthermore if <math|T> large enough and
    <math|k<rsub|1>,k<rsub|2>,k<rsub|3>\<in\>\<bbb-Z\><rsub|+><rsup|3>> and
    <math|k<rsub|1>+k<rsub|2>+k<rsub|3>=k>, while
    <math|<around*|\<langle\>|k<rsub|i>|\<rangle\>>\<geqslant\><around*|(|3T/32|)>\<times\>0.9>
    (recall that if <math|<around*|\<langle\>|k<rsub|i>|\<rangle\>>\<less\><around*|(|3T/32|)>\<times\>0.9><space|1em>and
    <math|s\<gtr\>3T/32> then <math|\<sigma\><rsub|s><around*|(|k<rsub|1>|)>=0>)
    we have \ <math|<around*|\<langle\>|k|\<rangle\>>\<geqslant\>T/8>. So for
    any <math|k> for which the integral is nonzero we have
    \ <math|\<rho\><rsub|T/2><around*|(|k|)>-\<rho\><rsub|T/8><around*|(|k|)>=1>
    (recall that <math|\<rho\>=1> on <math|B<around*|(|0,9/10|)>> and
    <math|\<rho\>=0> outside of <math|B<around*|(|0,1|)>>). This implies

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|<frac|1|T<rsup|2>><big|sum><rsub|k\<in\>\<bbb-Z\><rsub|+><rsup|3>><around*|(|\<rho\><rsub|T/2><around*|(|k|)>-\<rho\><rsub|T/8><around*|(|k|)>|)><big|sum><rsub|<tabular|<tformat|<table|<row|<cell|k<rsub|1>,k<rsub|2>,k<rsub|3>\<in\>\<bbb-Z\><rsub|+><rsup|3>>>|<row|<cell|k<rsub|1>+k<rsub|2>+k<rsub|3>=k>>>>>><big|int><rsup|T/8><rsub|3T/32><big|int><rsup|s<rsub|1>><rsub|3T/32><big|int><rsup|s<rsub|2>><rsub|3T/32><frac|\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>>>|<row|<cell|>|<cell|=>|<cell|<frac|1|T<rsup|2>><big|sum><rsub|<tabular|<tformat|<table|<row|<cell|k<rsub|1>,k<rsub|2>,k<rsub|3>\<in\>\<bbb-Z\><rsub|+><rsup|3>>>>>>><big|int><rsup|T/8><rsub|3T/32><big|int><rsup|s<rsub|1>><rsub|3T/32><big|int><rsup|s<rsub|2>><rsub|3T/32><frac|\<sigma\><rsup|2><rsub|s<rsub|1>><around*|(|k<rsub|1>|)>|<around*|\<langle\>|k<rsub|1>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|2>><around*|(|k<rsub|2>|)>|<around*|\<langle\>|k<rsub|2>|\<rangle\>><rsup|2>><frac|\<sigma\><rsup|2><rsub|s<rsub|3>><around*|(|k<rsub|3>|)>|<around*|\<langle\>|k<rsub|3>|\<rangle\>><rsup|2>>\<mathd\>s<rsub|1>\<mathd\>s<rsub|2>\<mathd\>s<rsub|3>>>|<row|<cell|>|<cell|\<gtrsim\>>|<cell|T>>>>
    </eqnarray*>

    \;
  </proof>

  Next we upgrade this bound to almost sure divergence.

  <\lemma>
    There exists a <math|\<delta\><rsub|0>\<gtr\>0> such that for any
    <math|\<delta\><rsub|0>\<geqslant\>\<delta\>\<gtr\>0>, <math|>,there
    exists a sequence <math|<around*|(|T<rsub|n>|)><rsub|n>> such that
    <math|\<bbb-P\>-almost surely>

    <\equation*>
      <frac|1|T<rsub|n><rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T<rsub|n>>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T<rsub|n>>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t\<rightarrow\>\<infty\>.
    </equation*>
  </lemma>

  <\proof>
    Define\ 

    <\equation*>
      G<rsub|T>\<assign\><frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t+
      sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|W<rsub|t>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>.
    </equation*>

    We will show that <math|e<rsup|-G<rsub|T>>\<rightarrow\>0> in
    <math|L<rsup|1><around*|(|\<bbb-P\>|)>>, which implies that there exists
    a subsequence <math|T<rsub|n>> such that
    <math|e<rsup|-G<rsub|T<rsub|n>>>\<rightarrow\>0> almost surely. From this
    our statement follows. By the Boué\UDupuis formula\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|>>|<row|<cell|-log\<bbb-E\><around*|[|e<rsup|-G<rsub|T>>|]>>|<cell|=>|<cell|inf<rsub|v\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>>16<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t><around*|\<llbracket\>|\<theta\><rsub|T><around*|(|<around*|(|W<rsub|t>+I<rsub|t><around*|(|v|)>|)><rsup|3>|)>|\<rrbracket\>>|)>J<rsub|t><around*|\<llbracket\>|<around*|(|W<rsub|t>+I<rsub|t><around*|(|v|)>|)><rsup|3>|\<rrbracket\>>\<mathd\>t+|\<nobracket\>>>>|<row|<cell|>|<cell|>|<cell|<space|2em><space|2em><around*|\<nobracket\>|+sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|W<rsub|t>+I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<frac|1|2><big|int><rsup|\<infty\>><rsub|0><around*|\<\|\|\>|v<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|>|<cell|=>|<cell|inf<rsub|v\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t|<right|.>>+>>|<row|<cell|>|<cell|>|<cell|<space|2em><space|2em>+<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|<around*|(|i,j|)>\<in\><around*|{|0,1,2,3|}><rsup|2>\<setminus\><around*|(|0,0|)>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>A<rsub|t><rsup|i>B<rsub|t><rsup|j>\<mathd\>t>>|<row|<cell|>|<cell|>|<cell|<space|2em><space|2em><around*|<left|.>|+sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|W<rsub|t>+I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<frac|1|2><big|int><rsup|\<infty\>><rsub|0><around*|\<\|\|\>|v<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|>|<cell|\<geqslant\>>|<cell|inf<rsub|v\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t|\<nobracket\>>>>|<row|<cell|>|<cell|>|<cell|<space|2em><space|2em>+<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|<around*|(|i,j|)>\<in\><around*|{|0,1,2,3|}><rsup|2>\<setminus\><around*|(|0,0|)>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>A<rsub|t><rsup|i>B<rsub|t><rsup|j>\<mathd\>t>>|<row|<cell|>|<cell|>|<cell|<space|2em><space|2em><around*|\<nobracket\>|+<frac|1|2>sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>-C
      sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|W<rsub|t>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<frac|1|2><big|int><rsup|\<infty\>><rsub|0><around*|\<\|\|\>|v<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>>>
    </eqnarray*>

    where where have used that <math|\<theta\><rsub|T>J<rsub|t>=0> for
    <math|t\<geqslant\>T> and introduced the notations, for
    <math|0\<leqslant\>i\<leqslant\>3>,

    <\equation*>
      A<rsub|t><rsup|i>\<assign\>4<choose|3|i>J<rsub|t>\<theta\><rsub|T><around*|(|<around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsub|t>|)><rsup|3-i>|\<rrbracket\>><around*|(|\<theta\><rsub|T>I<rsub|t><around*|(|v|)>|)><rsup|i>|)>,
    </equation*>

    and\ 

    <\equation*>
      B<rsup|i><rsub|t>\<assign\>4<choose|3|i>J<rsub|t><around*|(|<around*|\<llbracket\>|W<rsub|t><rsup|3-i>|\<rrbracket\>><around*|(|I<rsub|t>*<around*|(|v|)>|)><rsup|i>|)>.
    </equation*>

    Our aim now to prove that the last three terms are bounded below
    uniformly as <math|T\<rightarrow\>\<infty\>> (while we already know that
    the first one diverges). For <math|i\<in\><around*|{|1,2,3|}>>

    <\equation*>
      <around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>+<around*|\<\|\|\>|B<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-1+\<delta\>><around*|(|<around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+Q<rsub|t><around*|(|W|)>|)>
    </equation*>

    by Lemmas<nbsp><reference|analyticestimate2>
    and<nbsp><reference|analyticestimate4>. Here
    <math|Q<rsub|t><around*|(|W|)>> is a random variable only depending on
    <math|W> such that <math|sup<rsub|t>\<bbb-E\><around*|[|<around*|\||Q<rsub|t><around*|(|W|)>|\|><rsup|p>|]>\<less\>\<infty\>>
    for any <math|p\<less\>\<infty\>>. Then\ 

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|<around*|(|i,j|)>\<in\><around*|{|0,1,2,3|}><rsup|2>\<setminus\><around*|(|0,0|)>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\||A<rsub|t><rsup|i>B<rsub|t><rsup|j>|\|>\<mathd\>t>>|<row|<cell|\<leqslant\>>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|<around*|(|i,j|)>\<in\><around*|{|1,2,3|}><rsup|2>><big|int><rsup|T><rsub|0><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>+<around*|\<\|\|\>|B<rsub|t><rsup|j>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t>>|<row|<cell|>|<cell|+<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|i\<in\><around*|{|1,2,3|}>><big|int><rsup|T><rsub|0><around*|\<\|\|\>|A<rsub|t><rsup|0>|\<\|\|\>><rsub|L<rsup|2>>
      <around*|\<\|\|\>|B<rsub|t><rsup|i>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t+<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|i\<in\><around*|{|1,2,3|}>><big|int><rsup|T><rsub|0><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsub|L<rsup|2>>
      <around*|\<\|\|\>|B<rsub|t><rsup|0>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t.>>>>>
    </equation*>

    Now for the first term we obtain\ 

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|>|<cell|\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|<around*|(|i,j|)>\<in\><around*|{|1,2,3|}><rsup|2>><big|int><rsup|T><rsub|0><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>+<around*|\<\|\|\>|B<rsub|t><rsup|j>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|=>|<cell|\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>><big|sum><rsub|<around*|(|i,j|)>\<in\><around*|{|1,2,3|}><rsup|2>><big|int><rsup|T><rsub|0><around*|\<langle\>|t|\<rangle\>><rsup|-1+\<delta\>><around*|(|<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+Q<rsub|t><around*|(|W|)>|)>\<mathd\>t|]>>>|<row|<cell|=>|<cell|<frac|C|T<rsup|1-2\<delta\>>>\<bbb-E\><around*|[|sup<rsub|t><around*|(|<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>|)>|]>+<frac|C|T<rsup|1-2\<delta\>>>.>>>>>
    </equation*>

    For the second term we use that <math|<around*|\<\|\|\>|A<rsub|t><rsup|0>|\<\|\|\>><rsub|L<rsup|2>>\<leqslant\>Q<rsub|t><around*|(|W|)>>
    so\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|A<rsub|t><rsup|0>|\<\|\|\>><rsub|L<rsup|2>>
      <around*|\<\|\|\>|B<rsub|t><rsup|i>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><around*|\<langle\>|t|\<rangle\>><rsup|-1/2><around*|\<\|\|\>|A<rsub|t><rsup|0>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t+<big|int><rsup|T><rsub|0><around*|\<langle\>|t|\<rangle\>><rsup|1/2><around*|\<\|\|\>|B<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><around*|\<langle\>|t|\<rangle\>><rsup|-1/2><around*|\<\|\|\>|A<rsub|t><rsup|0>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|>|<cell|>|<cell|+<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><around*|\<langle\>|t|\<rangle\>><rsup|-1/2+\<delta\>><around*|(|<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+Q<rsub|t><around*|(|W|)>|)>\<mathd\>t|]>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<frac|C|T<rsup|1/2-2\<delta\>>>\<bbb-E\><around*|[|sup<rsub|t><around*|(|<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>|)>|]>+<frac|C|T<rsup|1/2-2\<delta\>>>>>>>
    </eqnarray*>

    Since <math|sup<rsub|t><around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>\<lesssim\><big|int><rsup|\<infty\>><rsub|0><around*|\<\|\|\>|v<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t>
    in total we obtain for <math|T> large enough. The third term is estimated
    analogously.

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|-log\<bbb-E\><around*|[|e<rsup|-G<rsub|T>>|]>>>|<row|<cell|>|<cell|\<geqslant\>>|<cell|inf<rsub|v\<in\>\<bbb-H\><rsub|a>>\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t+<around*|(|<frac|1|2>-<frac|C|T<rsup|1/2-2\<delta\>>>|)>sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|I<rsub|t><around*|(|v|)>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>|\<nobracket\>>>>|<row|<cell|>|<cell|>|<cell|<space|2em><around*|\<nobracket\>|-C
      sup<rsub|t\<less\>\<infty\>><around*|\<\|\|\>|W<rsub|t>|\<\|\|\>><rsup|K><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>>+<around*|(|<frac|1|2>-<frac|C|T<rsup|1/2-2\<delta\>>>|)><big|int><rsup|\<infty\>><rsub|0><around*|\<\|\|\>|v<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t-<frac|C|T<rsup|1/2-2\<delta\>>>|]>>>|<row|<cell|>|<cell|\<geqslant\>>|<cell|\<bbb-E\><around*|[|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t|]>-C\<rightarrow\>\<infty\>>>>>
    </eqnarray*>

    \;
  </proof>

  Next we prove an estimate which will help with the proof of the main
  theorem.\ 

  <\lemma>
    <label|Qcubeestimate>We have

    <\equation*>
      sup<rsub|T> \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>><frac|1|t<rsup|1+\<delta\>>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)><rsup|2>\<mathd\>t|]>\<less\>\<infty\>.
    </equation*>

    Furthermore, there exists a (deterministic) subsequence
    <math|<around*|(|T<rsub|n>|)><rsub|n>> such that \ 

    <\equation*>
      <frac|1|T<rsub|n><rsup|1/2+\<delta\>>><around*|\||<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T<rsub|n>>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T<rsub|n>>,3>\<mathd\>X<rsup|u><rsub|t>|\|>\<rightarrow\>0
    </equation*>

    <math|\<bbb-Q\><rsup|u>> almost surely.
  </lemma>

  <\proof>
    Recall that under <math|\<bbb-Q\><rsup|u>> we have
    <math|W<rsub|t>=W<rsub|t><rsup|u>+I<rsub|t><around*|(|u|)>> where
    <math|u> is defined above by <eqref|eq:drift> and
    <math|Law<rsub|\<bbb-Q\><rsup|u>><around*|(|W<rsup|u>|)>=Law<rsub|\<bbb-P\>><around*|(|W|)>>.
    With this in mind we compute\ 

    <\equation*>
      <big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><frac|1|t<rsup|1+\<delta\>>><around*|(|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|)><rsup|2>\<mathd\>t=<big|sum><rsub|i,j\<leqslant\>3><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><frac|1|t<rsup|1+\<delta\>>>A<rsub|t><rsup|i>A<rsub|t><rsup|j>\<mathd\>t,
    </equation*>

    where, as above,

    <\equation*>
      A<rsub|t><rsup|i>=4<choose|3|i>J<rsub|t>\<theta\><rsub|T><around*|(|<around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsup|u><rsub|t>|)><rsup|3-i>|\<rrbracket\>><around*|(|\<theta\><rsub|T>I<rsub|t><around*|(|u|)>|)><rsup|i>|)>.
    </equation*>

    By Lemmas<nbsp><reference|analyticestimate2>
    and<nbsp><reference|analyticestimate4> we have that
    <math|\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>|]>\<leqslant\>C>
    so the Cauchy\USchwartz inequality gives the result.
  </proof>

  <\theorem>
    There exists a sequence <math|<around*|(|T<rsub|n>|)><rsub|n>> such that,
    <math|\<bbb-Q\><rsup|u>> almost surely,

    <\equation*>
      <frac|1|T<rsup|1-\<delta\>><rsub|n>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T<rsub|n>>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>\<rightarrow\>-\<infty\>.
    </equation*>
  </theorem>

  <\proof>
    We have

    <\equation*>
      <big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>=<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsub|t>.
    </equation*>

    Now since <math|\<mathd\>X<rsub|t>=\<mathd\>X<rsub|t><rsup|u>+u<rsub|t>\<mathd\>t>
    we have

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsub|t>>>|<row|<cell|=>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsup|u><rsub|t>+<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>u<rsub|t>\<mathd\>t>>|<row|<cell|=>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>\<mathd\>X<rsup|u><rsub|t>-<frac|\<lambda\>|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t>>|<row|<cell|>|<cell|-<frac|\<lambda\>|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|<wide|T|\<bar\>>><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|u,2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>\<mathd\>t>>|<row|<cell|>|<cell|-<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t><around*|\<langle\>|D|\<rangle\>><rsup|-1/2><around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>\<mathd\>t.>>>>>
    </equation*>

    The first term goes to <math|0> <math|\<bbb-Q\><rsup|u>>-almost surely by
    Lemma<nbsp><reference|Qcubeestimate>. To analyze the third term we
    estimate

    <\equation>
      <tabular|<tformat|<table|<row|<cell|>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|<wide|T|\<bar\>>><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|u,2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>\<mathd\>t>>|<row|<cell|=>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|<wide|T|\<bar\>>><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|u,2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>\<mathd\>t>>|<row|<cell|\<leqslant\>>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|<wide|T|\<bar\>>><around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsub|L<rsup|2>><around*|\<\|\|\>|J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|u,2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t>>|<row|<cell|\<lesssim\>>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|<wide|T|\<bar\>>>t<rsup|-1/2+\<delta\>/2><around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsub|L<rsup|2>>
      <around*|\<\|\|\>|\<bbb-W\><rsub|t><rsup|u,2>|\<\|\|\>><rsub|<with|math-font|cal|C><rsup|-1-\<delta\>/2>><around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t>>|<row|<cell|\<leqslant\>>|<cell|T<rsup|-1/2-2\<delta\>><around*|(|<big|int><rsup|T><rsub|<wide|T|\<bar\>>><around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|)><rsup|1/2>>>|<row|<cell|>|<cell|<space|2em>\<times\>T<rsup|-1/2+2\<delta\>><around*|(|<big|int><rsup|T><rsub|<wide|T|\<bar\>>>t<rsup|-1+\<delta\>><around*|(|<around*|\<\|\|\>|\<bbb-W\><rsub|t><rsup|u,2>|\<\|\|\>><rsub|<with|math-font|cal|C><rsup|-1-\<delta\>/2>><around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsub|L<rsup|2>>|)><rsup|2>|)><rsup|1/2>>>>>><label|bcross1>
    </equation>

    By the computation from Lemma<nbsp><reference|Qcubeestimate> we have\ 

    <\equation*>
      \<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|T<rsup|-1/2-2\<delta\>><around*|(|<big|int><rsup|T><rsub|<wide|T|\<bar\>>><around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|)><rsup|1/2>|]>\<rightarrow\>0,
    </equation*>

    and <math|sup<rsub|t>\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|(|<around*|\<\|\|\>|\<bbb-W\><rsub|t><rsup|u,2>|\<\|\|\>><rsub|<with|math-font|cal|C><rsup|-1-\<delta\>/2>><around*|\<\|\|\>|I<rsub|t><around*|(|u|)>|\<\|\|\>><rsub|L<rsup|2>>|)><rsup|2>|]>\<less\>\<infty\>>,
    so <eqref|bcross1> converges to <math|0> in
    <math|L<rsup|1><around*|(|\<bbb-Q\><rsup|u>|)>>. For the fourth term we
    proceed in the same way:\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|<big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t><around*|\<langle\>|D|\<rangle\>><rsup|-1/2><around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>\<mathd\>t>>|<row|<cell|>|<cell|=>|<cell|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t><around*|\<langle\>|D|\<rangle\>><rsup|-1/2><around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>\<mathd\>t>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<big|int><rsup|T><rsub|0><around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsub|L<rsup|2>><around*|\<\|\|\>|J<rsub|t><around*|\<langle\>|D|\<rangle\>><rsup|-1/2><around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<big|int><rsup|T><rsub|0><around*|(|<around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsub|L<rsup|2>>|)>t<rsup|-2+\<delta\>><around*|\<\|\|\>|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>|\<\|\|\>><rsub|H<rsup|-\<delta\>>>\<mathd\>t>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<around*|(|<big|int><rsup|T><rsub|0>t<rsup|-2<around*|(|1-\<delta\>|)>><around*|(|<around*|\<\|\|\>|\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>|\<\|\|\>><rsub|L<rsup|2>>|)><rsup|2>|)><rsup|1/2><around*|(|<big|int><rsup|T><rsub|0>t<rsup|-2<around*|(|1-\<delta\>|)>><around*|\<\|\|\>|<around*|\<llbracket\>|<around*|(|<around*|\<langle\>|D|\<rangle\>><rsup|-1/2>W<rsup|u><rsub|t>|)><rsup|n>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|H<rsup|-\<delta\>>>|)><rsup|1/2>>>>>
    </eqnarray*>

    which is bounded in expectation uniformly in <math|T>, so the fourth term
    goes to <math|0> in <math|L<rsup|1><around*|(|\<bbb-Q\><rsup|u>|)>> as
    well. It remains to analyze the second term. Again introducing the
    notation\ 

    <\equation*>
      A<rsub|t><rsup|i>=4<choose|3|i>J<rsub|t>\<theta\><rsub|T><around*|(|<around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsup|u><rsub|t>|)><rsup|3-i>|\<rrbracket\>><around*|(|\<theta\><rsub|T>I<rsub|t><around*|(|u|)>|)><rsup|i>|)>,
    </equation*>

    <\equation*>
      \<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,u,3>=4<around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsup|u><rsub|t>|)><rsup|3>|\<rrbracket\>>,
    </equation*>

    we have

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|\<infty\>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t>>|<row|<cell|>|<cell|=>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,u,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t+<big|sum><rsub|1\<leqslant\>i\<leqslant\>3><frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>A<rsub|t><rsup|i>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t.>>>>
    </eqnarray*>

    Now observe that\ 

    <\equation*>
      <frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,u,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t
      <rsub|> <rsub|\<bbb-Q\><rsup|u>>\<sim\><rsub|\<bbb-P\><space|1em>><frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|3>\<mathd\>t<rsub|>,
    </equation*>

    so the <math|limsup>of this is <math|\<infty\>> almost surely. To
    estimate the sum we again observe that for <math|i\<geqslant\>3>
    <math|\<bbb-E\><rsub|\<bbb-Q\><rsup|u>><around*|[|<around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>|]>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-1+\<delta\>>>
    and by Young's inequality\ 

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>A<rsub|t><rsup|i>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t>|<cell|\<leqslant\>>|<cell|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsub|L<rsup|2>><around*|\<\|\|\>|J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|1/3><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsub|L<rsup|2>><around*|\<langle\>|t|\<rangle\>><rsup|-1/3><around*|\<\|\|\>|J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>|\<\|\|\>><rsub|L<rsup|2>>\<mathd\>t>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|2/3><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>+<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|-2/3><around*|\<\|\|\>|J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t.>>>>>
    </equation*>

    Taking expectation we obtain\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>A<rsub|t><rsup|i>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t|]>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|2/3><around*|\<\|\|\>|A<rsub|t><rsup|i>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>|]>+<frac|1|T<rsup|1-\<delta\>>>\<bbb-E\><around*|[|<big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|-2/3><around*|\<\|\|\>|J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t|]>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|-1/3+\<delta\>>+<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|\<langle\>|t|\<rangle\>><rsup|-2/3>\<mathd\>t\<rightarrow\>0>>>>
    </eqnarray*>

    We have deduced that\ 

    <\equation*>
      <frac|1|T<rsup|1-\<delta\>>><big|int><rsub|\<Lambda\>><around*|\<llbracket\>|<around*|(|\<theta\><rsub|T>W<rsub|\<infty\>>|)><rsup|4>|\<rrbracket\>>=-<frac|1|T<rsup|1-\<delta\>>><big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T>,u,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t<rsub|>+R<rsub|T>,
    </equation*>

    where <math|R<rsub|T>\<rightarrow\>0> in
    <math|L<rsup|1><around*|(|\<bbb-Q\><rsup|u>|)>>. We can conclude by
    selecting a subsequence <math|<around*|(|T<rsub|n>|)><rsub|n>> such that\ 

    <\equation*>
      <frac|1|T<rsub|n><rsup|1-\<delta\>>><big|int><rsup|T<rsub|n>><rsub|0><big|int><rsub|\<Lambda\>>\<theta\><rsub|T>J<rsub|t>\<bbb-W\><rsub|t><rsup|\<theta\><rsub|T<rsub|n>>,u,3>J<rsub|t>\<bbb-W\><rsub|t><rsup|u,3>\<mathd\>t\<rightarrow\>\<infty\>
    </equation*>

    <math|\<bbb-Q\><rsup|u>>-almost surely and
    <math|R<rsub|T<rsub|n>>\<rightarrow\>0>, <math|\<bbb-Q\><rsup|u>>-almost
    surely.
  </proof>

  <section|Some analytic estimates ><label|sec:analytic>

  We collect in this final section various technical estimates needed to
  complete the proof of Lemma<nbsp><reference|lemma:estimateremainder>.

  <\proposition>
    <label|FractionalLeibniz>Let <math|1\<less\>p\<less\>\<infty\>> and
    <math|p<rsub|1>,p<rsub|2>,p<rsub|1><rprime|'>,p<rsub|2><rprime|'>\<gtr\>1>
    such that <math|<frac|1|p<rsub|1>>+<frac|1|p<rsub|2>>=<frac|1|p<rsub|1><rprime|'>>+<frac|1|p<rsub|2><rprime|'>>=<frac|1|p>>.
    Then for every <math|s,\<alpha\>\<geqslant\>0> \ 

    <\equation*>
      <around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|s><around*|(|f
      g|)>|\<\|\|\>><rsub|L<rsup|p>>\<lesssim\><around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|s+\<alpha\>>f|\<\|\|\>><rsub|L<rsup|p<rsub|2>>>
      <around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-\<alpha\>>g|\<\|\|\>><rsub|L<rsup|p<rsub|1>>>+<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|s+\<alpha\>>g|\<\|\|\>><rsub|L<rsup|p<rsub|1><rprime|'>>>
      <around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-\<alpha\>>f|\<\|\|\>><rsub|L<rsup|p<rsub|2><rprime|'>>>.
    </equation*>
  </proposition>

  <\proof>
    See<nbsp><cite|gulisashvili_exact_1996>.
  </proof>

  <\lemma>
    There exists <math|\<varepsilon\>\<gtr\>0,n\<in\>\<bbb-N\>> such that for
    any <math|\<delta\>\<gtr\>0> there exists
    <math|C<rsub|\<delta\>>\<less\>\<infty\>> for which the following
    inequality holds for any <math|\<phi\>\<in\>H<rsup|1><around*|(|\<Lambda\>|)>>
    <label|L4estimate>

    <\equation*>
      <around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|4+\<varepsilon\>><rsub|L<rsup|4>>\<leqslant\>C<around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|n+1><rsub|W<rsup|-1/2,n+1>>+\<delta\><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+C<rsub|\<delta\>>.
    </equation*>
  </lemma>

  <\proof>
    \;

    <\eqnarray*>
      <tformat|<table|<row|<cell|<big|int>\<phi\><rsup|4>\<mathd\>x>|<cell|\<leqslant\>>|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsub|L<rsup|8>><around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|1/2>\<phi\><rsup|3>|\<\|\|\>><rsub|L<rsup|8/7>>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsub|L<rsup|8>><around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|1/2>\<phi\>|\<\|\|\>><rsub|L<rsup|8/3>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|2><rsub|L<rsup|4>>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsub|L<rsup|8>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|1/2><rsub|H<rsup|1>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|5/2><rsub|L<rsup|4>>>>>>
    </eqnarray*>

    So

    <\eqnarray*>
      <tformat|<table|<row|<cell|<around*|(|<around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)><rsup|21/20>>|<cell|\<leqslant\>>|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsup|21/20><rsub|L<rsup|8>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|21/40><rsub|H<rsup|1>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|104/40><rsub|L<rsup|4>>>>>>
    </eqnarray*>

    and applying Young's inequality with the exponents
    <math|<around*|(|32,32/9,32/22|)>>, we obtain

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsup|21/20><rsub|L<rsup|8>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|21/40><rsub|H<rsup|1>><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|104/40><rsub|L<rsup|4>>>|<cell|\<leqslant\>>|<cell|C<rsub|\<delta\>><around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsup|168/5><rsub|L<rsup|8>>+\<delta\><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|16/9><rsub|H<rsup|1>>+\<delta\><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|208/55><rsub|L<rsup|4>>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|-1/2>\<phi\>|\<\|\|\>><rsup|34><rsub|L<rsup|8>>+\<delta\><around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+\<delta\><around*|(|<around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)><rsup|21/20>+C<rsub|\<delta\>>>>>>>
    </equation*>

    and subtracting <math|\<delta\><around*|(|<around*|\<\|\|\>|\<phi\>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)><rsup|21/20>>
    on both sides of the inequality gives the result.\ 
  </proof>

  <\lemma>
    The following estimates hold with <math|\<varepsilon\>\<gtr\>0> small
    enough<label|analyticestimate1>

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>\<succ\><around*|(|1-\<theta\><rsub|t>|)>I<rsub|t><around*|(|w|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><frac|1|<around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>>><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n><rsub|W<rsup|-1/2,n+1>>+<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>|)>
    </equation*>

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>\<circ\>I<rsub|t><around*|(|w|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><frac|1|<around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>>><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n><rsub|W<rsup|-1/2,n+1>>+<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>|)>
    </equation*>

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>\<prec\>I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><frac|1|<around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>>><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n><rsub|W<rsup|-1/2,n+1>>+<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>|)>
    </equation*>
  </lemma>

  <\proof>
    We observe that since <math|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>>
    is spectrally supported in a ball or radius <math|\<thicksim\>t>

    <\equation*>
      <around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsub|<value|VV><rsup|-1+\<varepsilon\>>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|2\<varepsilon\>><around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>
      .
    </equation*>

    For the first estimate we know that <math|<around*|(|1-\<theta\><rsub|t>|)>I<rsub|t><around*|(|w|)>>
    is supported in an annulus of radius <math|\<sim\>t>, so
    <math|<around*|\<\|\|\>|<around*|(|1-\<theta\><rsub|t>|)>I<rsub|t><around*|(|w|)>|\<\|\|\>><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-1+\<varepsilon\>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsub|H<rsup|1-\<varepsilon\>>>>
    and furthermore by interpolation <math|<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsub|H<rsup|1-\<varepsilon\>>>\<lesssim\><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|1-\<varepsilon\>><rsub|H<rsup|1>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|\<varepsilon\>><rsub|L<rsup|2>>\<lesssim\><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|1-\<varepsilon\>><rsub|H<rsup|1>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|\<varepsilon\>><rsub|L<rsup|4>>>.
    By definition <math|<around*|\<langle\>|t|\<rangle\>><rsup|1/2>J<rsub|t>>
    is a uniformly bounded Fourier multiplier regularizing by 1, and putting
    everything together, by paraproduct estimates

    <\eqnarray*>
      <tformat|<table|<row|<cell|<around*|\<\|\|\>|J<rsub|t><around*|(|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>\<succ\><around*|(|1-\<theta\><rsub|t>|)>I<rsub|t><around*|(|w|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-1><around*|\<langle\>|t|\<rangle\>><rsup|2\<varepsilon\>><around*|\<langle\>|t|\<rangle\>><rsup|-2+2\<varepsilon\>><around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1-\<varepsilon\>>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-1><around*|\<langle\>|t|\<rangle\>><rsup|2\<varepsilon\>><around*|\<langle\>|t|\<rangle\>><rsup|-2+2\<varepsilon\>><around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2-2\<varepsilon\>><rsub|H<rsup|1>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2\<varepsilon\>><rsub|L<rsup|4>>>>|<row|<cell|<small|<around*|(|\<varepsilon\>=2/7|)>><space|2em>>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|(|<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|14><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>|)>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w|\<\|\|\>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n><rsub|W<rsup|-1/2,n+1>>+<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|14><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>|)>>>>>
    </eqnarray*>

    For the second term in addition observe that the function
    <math|<around*|\<langle\>|t|\<rangle\>><rsup|1/2>J<rsub|t>> is spectrally
    supported in an annulus of radius<nbsp><math|\<sim\>t>, and regularizes
    by <math|1> so again by estimates for the resonant product\ 

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|J<rsub|t><around*|(|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>\<circ\>I<rsub|t><around*|(|w|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3><around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1+2\<varepsilon\>>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1-\<varepsilon\>>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3><around*|\<langle\>|t|\<rangle\>><rsup|6\<varepsilon\>><around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1-\<varepsilon\>>>>>>>>
    </equation*>

    For the third estimate again applying paraproduct estimates and the
    properties of <math|J>,

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|<around*|\<llbracket\>|W<rsub|s><rsup|2>|\<rrbracket\>>\<prec\>I<rsub|t><around*|(|w|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3+4\<varepsilon\>><around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|s><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1-\<varepsilon\>>><rsub|>.
    </equation*>

    Now, the claim follows from interpolation and Young's inequality

    <\eqnarray*>
      <tformat|<table|<row|<cell|>|<cell|>|<cell|<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1-\<varepsilon\>>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2-2\<varepsilon\>><rsub|H<rsup|1>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2\<varepsilon\>><rsub|L<rsup|4>>>>|<row|<cell|<small|<around*|(|\<varepsilon\>=2/7|)>><space|2em>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|14><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|(|<big|int><rsup|t><rsub|0><around*|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<mathd\>s+<around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|n><rsub|W<rsup|-1/2,n+1>>+<around*|\<\|\|\>|<around*|\<llbracket\>|W<rsub|t><rsup|2>|\<rrbracket\>>|\<\|\|\>><rsup|14><rsub|<value|VV><rsup|-1-\<varepsilon\>>><rsub|>|)>.>>>>
    </eqnarray*>

    \;
  </proof>

  <\lemma>
    <label|analyticestimate2>Let <math|f\<in\>C<around*|(|<around*|[|0,\<infty\>|]>,<value|VV><rsup|-1/2-\<varepsilon\>>|)>>
    and <math|g\<in\>C<around*|(|<around*|[|0,\<infty\>|]>,H<rsup|1>|)>> such
    that <math|f<rsub|t>,g<rsub|t>> have spectral support in a ball of radius
    proportional to <math|t>. There exists <math|n\<in\>\<bbb-N\>> such that
    the following estimates hold:

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|f<rsub|t>g<rsub|t><rsup|2>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|<value|VV><rsup|-1/2-\<delta\>>><rsup|2><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>,
    </equation*>

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|f<rsub|t>g<rsub|t><rsup|2>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|(|<around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<delta\>>><rsub|>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|W<rsup|-1/2,n>><rsup|n>|)>,
    </equation*>

    and\ 

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|g<rsub|t><rsup|3>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|(|<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|H<rsup|1>><rsup|2>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|W<rsup|-1/2,n>><rsup|n>|)>.
    </equation*>
  </lemma>

  <\proof>
    By the spectral properties of <math|J<rsub|t>>,

    <\equation*>
      <around*|\<\|\|\>|J<rsub|t><around*|(|f<rsub|t>g<rsub|t><rsup|2>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|\<infty\>>><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|4><rsub|L<rsup|4>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1/2-\<delta\>>><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|L<rsup|4>><rsup|4>.
    </equation*>

    Applying Young's inequality with exponents
    <math|<around*|(|<frac|n|2>,<frac|n/2|<around*|(|n/2-1|)>>|)>> with
    <math|n> such that <math|<frac|2n|<around*|(|n/2-1|)>>\<leqslant\>4+\<varepsilon\>>
    where <math|\<varepsilon\>> is chosen as in Lemma <reference|L4estimate>
    we have\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1/2-\<delta\>>><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|L<rsup|4>><rsup|4>>|<cell|\<leqslant\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|(|<around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<delta\>>>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|L<rsup|4>><rsup|4+\<varepsilon\>>|)>>>|<row|<cell|>|<cell|\<leqslant\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3/2><around*|(|<around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<delta\>>>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|W<rsup|-1/2,n>><rsup|n>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>|)>>>>>
    </eqnarray*>

    Now the second estimate follows from <math|chosing > <math|n> large
    enough (depending on <math|\<delta\>>) and using Besov embedding after
    taking <math|f=g>.
  </proof>

  <\lemma>
    The following estimates hold<label|analyticestimate3>

    <\equation*>
      <around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>><around*|\<\|\|\>|J<rsub|s><around*|(|W<rsub|s>I<rsub|t><around*|(|w|)>\<succ\><rsub|>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4+\<varepsilon\>><rsub|L<rsup|4>>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u|)>|\<\|\|\>><rsup|4><rsub|L<rsup|4>><rsub|>+<around*|\<\|\|\>|W<rsub|t>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>,
    </equation*>

    <\equation*>
      <around*|\<langle\>|t|\<rangle\>><rsup|1+\<varepsilon\>><around*|\<\|\|\>|J<rsub|s><around*|(|<around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>\<succ\>I<rsup|\<flat\>><rsub|s><around*|(|u|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4+\<varepsilon\>><rsub|L<rsup|4>>+<around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u|)>|\<\|\|\>><rsup|n><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>.
    </equation*>
  </lemma>

  <\proof>
    For the first estimate we again use the spectral properties of
    <math|W,I,> and <math|J> and obtain by paraproduct estimate\ 

    <\eqnarray*>
      <tformat|<table|<row|<cell|<around*|\<\|\|\>|J<rsub|s><around*|(|W<rsub|t>I<rsub|t><around*|(|w|)>\<succ\><rsub|>I<rsup|\<flat\>><rsub|t><around*|(|u|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3><around*|\<\|\|\>|W<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|\<infty\>>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsub|L<rsup|4>><rsup|2><around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u|)>|\<\|\|\>><rsup|2><rsub|L<rsup|4>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-3><around*|\<langle\>|t|\<rangle\>><rsup|1+4\<varepsilon\>><around*|\<\|\|\>|W<rsub|t>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsub|L<rsup|4>><rsup|2><around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u|)>|\<\|\|\>><rsup|2><rsub|L<rsup|4>>>>>>
    </eqnarray*>

    and the claim follows by Young's inequality. For the second\ 

    <\equation*>
      <around*|\<\|\|\>|J<rsub|s><around*|(|<around*|(|I<rsub|s><around*|(|w|)>|)><rsup|2>\<succ\>I<rsup|\<flat\>><rsub|s><around*|(|u|)>|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|2-2\<varepsilon\>><around*|\<\|\|\>|<around*|(|I<rsub|s><around*|(|w|)>|)>|\<\|\|\>><rsub|L<rsup|4>><rsup|4><around*|\<\|\|\>|I<rsup|\<flat\>><rsub|t><around*|(|u|)>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1/2-\<varepsilon\>>><rsub|>,
    </equation*>

    and the claim follows again by Young's inequality.\ 
  </proof>

  <\lemma>
    <label|analyticestimate4>Let <math|f<rsub|t>\<in\>C<around*|(|<around*|[|0,\<infty\>|]>,<value|VV><rsup|-1/2-\<delta\>>|)>>
    and <math|g<rsub|t>\<in\>C<around*|(|<around*|[|0,\<infty\>|]>,H<rsup|1>|)>>
    such that <math|f<rsub|t>,g<rsub|t>> have spectral support in a ball of
    radius proportional to <math|t>. Then the following estimates hold\ 

    <\equation*>
      <around*|\<\|\|\>|<around*|(|J<rsub|t><around*|(|f<rsub|t>g<rsub|t>|)>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-1+2\<delta\>><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<delta\>>><rsup|2><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>
    </equation*>

    <\equation*>
      <around*|\<\|\|\>|<around*|(|J<rsub|t><around*|(|f<rsub|t>g<rsub|t>|)>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-1+2\<delta\>><around*|(|<around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsup|8><rsub|<value|VV><rsup|-1-\<delta\>>>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|4><rsub|H<rsup|-1>>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|H<rsup|1>>|)>
    </equation*>
  </lemma>

  <\proof>
    \;

    <\equation*>
      <around*|\<\|\|\>|<around*|(|J<rsub|t><around*|(|f<rsub|t>g<rsub|t>|)>|)>|\<\|\|\>><rsub|L<rsup|2>><rsup|2>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-3><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|L<rsup|\<infty\>>><rsup|2><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<lesssim\><around*|\<langle\>|t|\<rangle\>><rsup|-1+2\<delta\>><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<delta\>>><rsup|2><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>.
    </equation*>

    This proves the first estimate. For the second we continue

    <\eqnarray*>
      <tformat|<table|<row|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-1+2\<delta\>><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<delta\>>><rsup|2><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-1+2\<delta\>><around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<delta\>>><rsup|2><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|H<rsup|1>><around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsub|H<rsup|-1>>>>|<row|<cell|>|<cell|\<lesssim\>>|<cell|<around*|\<langle\>|t|\<rangle\>><rsup|-1+2\<delta\>><around*|(|<around*|\<\|\|\>|f<rsub|t>|\<\|\|\>><rsub|<value|VV><rsup|-1-\<delta\>>><rsup|8>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|4><rsub|H<rsup|-1>>+<around*|\<\|\|\>|g<rsub|t>|\<\|\|\>><rsup|2><rsub|H<rsup|-1>>|)>.>>>>
    </eqnarray*>

    \;
  </proof>

  <\lemma>
    <label|analyticestimate5>It holds

    <\equation*>
      <big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|(|J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|w|)>|)>|)><rsup|2>\<lesssim\>T<rsup|3\<delta\>><around*|(|sup<rsub|t><around*|\<\|\|\>|\<bbb-W\><rsub|t><rsup|2>|\<\|\|\>><rsup|2><rsub|<value|VV><rsup|-1-\<delta\>>>|)><around*|(|sup<rsub|t><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>|)>,
    </equation*>

    and

    <\equation*>
      <big|int><rsup|T><rsub|0><big|int><rsub|\<Lambda\>><around*|(|J<rsub|t><around*|(|\<bbb-W\><rsub|t><rsup|2>\<succ\>I<rsup|\<flat\>><rsub|t><around*|(|w|)>|)>|)><rsup|2>\<lesssim\>T<rsup|3\<delta\>><around*|(|sup<rsub|t><around*|\<\|\|\>|I<rsub|t><around*|(|w|)>|\<\|\|\>><rsup|4><rsub|H<rsup|-1>>+<big|int><rsup|T><rsub|0><around*|\<\|\|\>|w<rsub|t>|\<\|\|\>><rsup|2><rsub|L<rsup|2>>\<mathd\>t<rsub|>+sup<rsub|t><around*|\<\|\|\>|\<bbb-W\><rsub|t><rsup|2>|\<\|\|\>><rsup|8><rsub|<value|VV><rsup|-1-\<delta\>>>|)>.
    </equation*>

    \;
  </lemma>

  <\proof>
    This follows in the same fashion as Lemma<nbsp><reference|analyticestimate4>
    .
  </proof>

  <appendix|Besov spaces and paraproducts><label|sec:appendix-para>

  In this section we will recall some well known results about Besov spaces,
  embeddings, Fourier multipliers and paraproducts. The reader can find full
  details and proofs in<nbsp><cite|bahouri_fourier_2011|gubinelli_paracontrolled_2015>.\ 

  \;

  First recall the definition of Littlewood\UPaley blocks. Let
  <math|\<chi\>,\<varphi\>> be smooth radial functions
  <math|\<bbb-R\><rsup|d>\<rightarrow\>\<bbb-R\>> such that\ 

  <\itemize>
    <item><math|supp\<chi\>\<subseteq\>B<around*|(|0,R|)>>,
    <math|supp\<varphi\>\<subseteq\>B<around*|(|0,2R|)>\<setminus\>B<around*|(|0,R|)>>;

    <item><math|0\<leqslant\>\<chi\>,\<varphi\>\<leqslant\>1>,
    <math|\<chi\><around*|(|\<xi\>|)>+<big|sum><rsub|j\<geq\>0>\<varphi\><around*|(|2<rsup|-j>\<xi\>|)>=1>
    for any <math|\<xi\>\<in\>\<bbb-R\><rsup|d>>;

    <item><math|supp \<varphi\><around*|(|2<rsup|-j>\<cdot\>|)>\<cap\>supp
    \<varphi\><around*|(|2<rsup|-i>\<cdot\>|)>=\<varnothing\>> if
    <math|<around*|\||i-j|\|>\<gtr\>1>.
  </itemize>

  Introduce the notations <math|\<varphi\><rsub|-1>=\<chi\>>,
  <math|\<varphi\><rsub|j>=\<varphi\><around*|(|2<rsup|-j>\<cdot\>|)>> for
  <math|j\<geqslant\>0>. For any <math|f\<in\><value|CS><rprime|'><around*|(|\<Lambda\>|)>>
  we define the operators <math|\<Delta\><rsub|j>f\<assign\><with|math-font|cal|<value|CF>><rsup|-1><rsub|\<xi\>><around*|(|\<varphi\><rsub|j><around*|(|\<xi\>|)><wide|f|^><around*|(|\<xi\>|)>|)>>,
  <math|j\<geqslant\>-1>.

  <\definition>
    Let <math|s\<in\>\<bbb-R\>,p,q\<in\><around*|[|1,\<infty\>|]>>. For a
    Schwarz distribution <math|f\<in\><value|CS><rprime|'><around*|(|\<Lambda\>|)>>
    define the norm\ 

    <\equation*>
      <around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,q><rsup|s>>\<assign\><around*|\<\|\|\>|<around*|(|2<rsup|j
      s><around*|\<\|\|\>|\<Delta\><rsub|j>f|\<\|\|\>><rsub|L<rsup|p>>|)><rsub|j\<geqslant\>-1>|\<\|\|\>><rsub|\<ell\><rsup|q>>.
    </equation*>

    Then the space <math|B<rsup|s><rsub|p,q>> is the closure of Schwarz
    distributions under this norm. We denote
    <math|<VV><rsup|\<alpha\>>=B<rsub|\<infty\>,\<infty\>><rsup|\<alpha\>>>
    the Besov\UHölder space and <math|H<rsup|\<alpha\>>=B<rsup|\<alpha\>><rsub|2,2>>
    the Sobolev spaces. \ 
  </definition>

  <\proposition>
    <label|besovembedding>Let <math|1\<leqslant\>p<rsub|1>\<leqslant\>p<rsub|2>\<leqslant\>\<infty\>>
    and <math|1\<leqslant\>q<rsub|1>\<leqslant\>q<rsub|2>\<leqslant\>\<infty\>>.
    Then <math|B<rsup|s><rsub|p<rsub|1>,q<rsub|1>>> is continuously embedded
    in <math|B<rsub|p<rsub|2>,q<rsub|2>><rsup|s-d<around*|(|<frac|1|p<rsub|1>>-<frac|1|p<rsub|2>>|)>>>.
  </proposition>

  <\proposition>
    <label|compactembedding>For any <math|s<rsub|1>,s<rsub|2>\<in\>\<bbb-R\>>
    such that <math|s<rsub|1>\<less\>s<rsub|2>>, any
    <math|p,q\<in\><around*|[|1,\<infty\>|]>> the Besov space
    <math|B<rsup|s<rsub|1>><rsub|p,q<rsub|>>> is compactly embedded into
    <math|B<rsup|s<rsub|2>><rsub|p,q>>.
  </proposition>

  <\definition>
    Let <math|f,g\<in\><with|math-font|cal|<value|CS>><around*|(|\<Lambda\>|)>>.
    We define the paraproducs

    <\equation*>
      f\<succ\>g\<assign\><big|sum><rsub|j\<less\>i-1>\<Delta\><rsub|i>f\<Delta\><rsub|j>g,<space|2em><text|and><space|2em>f\<prec\>g\<assign\><big|sum><rsub|j\<gtr\>i+1>\<Delta\><rsub|i>f\<Delta\><rsub|j>g=g\<succ\>f.
    </equation*>

    Moreover we introduce the resonant product

    <\equation*>
      f\<circ\>g\<assign\><big|sum><rsub|<around*|\||i-j|\|>\<leqslant\>1>\<Delta\><rsub|i>f\<Delta\><rsub|j>g.
    </equation*>

    Then <math|f*g=f\<prec\>g+f\<circ\>g+f\<succ\>g>.
  </definition>

  <\proposition>
    <label|paraproductestimate>Let <math|\<alpha\>\<less\>0,\<beta\>\<in\>\<bbb-R\>>.
    For <math|f,g\<in\><value|CS><around*|(|\<Lambda\>|)>> we have the
    estimates\ 

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|f\<succ\>g|\<\|\|\>><rsub|H<rsup|\<beta\>-\<delta\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|L<rsup|2>>,>|<cell|>|<cell|<around*|\<\|\|\>|f\<succ\>g|\<\|\|\>><rsub|<VV><rsup|\<beta\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|L<rsup|\<infty\>>>,>>|<row|<cell|<around*|\<\|\|\>|f\<succ\>g|\<\|\|\>><rsub|H<rsup|\<beta\>-\<alpha\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|H<rsup|\<alpha\>>>,>|<cell|<space|2em>>|<cell|<around*|\<\|\|\>|f\<succ\>g|\<\|\|\>><rsub|<VV><rsup|\<beta\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|<VV><rsup|\<alpha\>>>.>>>>>
    </equation*>

    \;

    Let <math|\<alpha\>,\<beta\>\<in\>\<bbb-R\>> such that
    <math|\<alpha\>+\<beta\>\<gtr\>0>. Then

    <\equation*>
      <tabular|<tformat|<table|<row|<cell|<around*|\<\|\|\>|f\<circ\>g|\<\|\|\>><rsub|H<rsup|\<alpha\>+\<beta\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|H<rsup|\<alpha\>>>,>|<cell|<space|2em>>|<cell|<around*|\<\|\|\>|f\<circ\>g|\<\|\|\>><rsub|<VV><rsup|\<alpha\>+\<beta\>>>>|<cell|\<lesssim\>>|<cell|<around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|<VV><rsup|\<alpha\>>>.>>>>>
    </equation*>

    By density the paraproduct and resonant product also extend to bilinear
    operators on the respective spaces.\ 
  </proposition>

  <\proposition>
    <label|commutatorestimate>Let <math|\<alpha\>\<in\><around*|(|0,1|)>>
    <math|\<beta\>,\<gamma\>\<in\>\<bbb-R\>> such that
    <math|\<beta\>+\<gamma\>\<less\>0>, <math|\<alpha\>+\<beta\>+\<gamma\>\<gtr\>0>.
    Then for <math|f,g,h\<in\><value|CS>>, and for any
    <math|\<delta\>\<gtr\>0>,

    <\equation*>
      <around*|\<\|\|\>|<around*|(|f\<succ\>g|)>\<circ\>h-g<around*|(|f\<circ\>h|)>|\<\|\|\>><rsub|H<rsup|\<alpha\>+\<beta\>+\<gamma\>-\<delta\>>>\<lesssim\><around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<gamma\>>><around*|\<\|\|\>|h|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|H<rsup|\<alpha\>>>,
    </equation*>

    <\equation*>
      <around*|\<\|\|\>|<around*|(|f\<succ\>g|)>\<circ\>h-g<around*|(|f\<circ\>h|)>|\<\|\|\>><rsub|<VV><rsup|\<alpha\>+\<beta\>+\<gamma\>>>\<lesssim\><around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<gamma\>>><around*|\<\|\|\>|h|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|<VV><rsup|\<alpha\>>>.
    </equation*>
  </proposition>

  <\proposition>
    <label|adjointparaproduct>Assume <math|f\<in\><VV><rsup|\<alpha\>>,g\<in\>H<rsup|\<beta\>>,h\<in\>H<rsup|\<gamma\>>>
    and <math|\<alpha\>+\<beta\>+\<gamma\>=0>. Then\ 

    <\equation*>
      <big|int><rsub|\<bbb-T\><rsup|d>><around*|[|<around*|(|f\<succ\>g|)>h-<around*|(|f\<circ\>h|)>g|]>\<lesssim\><around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<alpha\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|H<rsup|\<beta\>>><around*|\<\|\|\>|h|\<\|\|\>><rsub|H<rsup|\<gamma\>>>.
    </equation*>
  </proposition>

  <\remark>
    Proposition<nbsp><reference|adjointparaproduct> is not proven in the
    above references but is quite easy and the reader can fill out a proof.
  </remark>

  <\definition>
    A smooth function <math|\<eta\>:\<bbb-R\><rsup|d>\<rightarrow\>\<bbb-R\>>
    is said to be an <math|S<rsup|m>>-multiplier if for every multi-index
    <math|\<alpha\>> there exists a constant <math|C<rsub|\<alpha\>>> such
    that\ 

    <\equation>
      <space|1em><around*|\||<frac|\<partial\><rsup|\<alpha\>>|\<partial\>\<xi\><rsup|\<alpha\>>>f<around*|(|\<xi\>|)>|\|>\<lesssim\><rsub|\<alpha\>><around*|(|1+<around*|\||\<xi\>|\|>|)><rsup|m-<around*|\||\<alpha\>|\|>>,<space|2em><space|2em><space|2em>\<forall\>\<xi\>\<in\>\<bbb-R\><rsup|d>.<label|symbolinq>
    </equation>

    We say that a family <math|<around*|(|\<eta\><rsub|t>|)><rsub|t\<geqslant\>0>>
    is a uniform <math|S<rsup|m>>-multiplier if <eqref|symbolinq> is
    satisfied for every <math|\<eta\><rsub|t>> with <math|C<rsub|\<alpha\>>>
    independent of <math|t\<geqslant\>0>.\ 
  </definition>

  <\proposition>
    <label|multiplierestimate>Let <math|\<eta\>> be an
    <math|S<rsup|m>>-multiplier, <math|s\<in\>\<bbb-R\>>,
    <math|p,q\<in\><around*|[|1,\<infty\>|]>>, and
    <math|f\<in\>B<rsub|p,q><rsup|s><around*|(|\<bbb-T\><rsup|d>|)>>, then\ 

    <\equation*>
      <around*|\<\|\|\>|\<eta\><around*|(|\<mathD\>|)>f|\<\|\|\>><rsub|B<rsub|p,q><rsup|s-m>>\<lesssim\><around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,q><rsup|s>>.
    </equation*>

    Furthermore the constant depends only on <math|s,p,q,d> and the constants
    <math|C<rsub|\<alpha\>>> in <eqref|symbolinq>.
  </proposition>

  <\proposition>
    <label|paraproductleibniz>Assume <math|m\<leqslant\>0>,
    <math|\<alpha\>\<in\><around*|(|0,1|)>,\<beta\>\<in\>\<bbb-R\>>. Let
    <math|\<eta\>> be an <math|S<rsup|m>>-multiplier,
    <math|f\<in\><VV><rsup|\<beta\>>>, <math|g\<in\>H<rsup|\<alpha\>>>. Then
    for any <math|\<delta\>\<gtr\>0>.

    <\equation*>
      <around*|\<\|\|\>|\<eta\><around*|(|\<mathD\>|)><around*|(|f\<succ\>g|)>-<around*|(|\<eta\><around*|(|\<mathD\>|)>f\<succ\>g|)>|\<\|\|\>><rsub|H<rsup|\<alpha\>+\<beta\>-m-\<delta\>>>\<lesssim\><around*|\<\|\|\>|f|\<\|\|\>><rsub|<VV><rsup|\<beta\>>><around*|\<\|\|\>|g|\<\|\|\>><rsub|H<rsup|\<alpha\>>>.
    </equation*>

    Again the constant depends only on <math|\<alpha\>,\<beta\>,\<delta\>>
    and the constants in<nbsp><eqref|symbolinq>.
  </proposition>

  <\proposition>
    <label|besovembedding>Let <math|\<delta\>\<gtr\>0>.We have for any
    <math|q<rsub|1>,q<rsub|2>\<in\><around*|[|1,\<infty\>|]>,q<rsub|1>\<less\>q<rsub|2>>

    <\equation*>
      <around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,q<rsub|2>><rsup|s>>\<leqslant\><around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,q<rsub|1>><rsup|s>>\<leqslant\><around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,\<infty\>><rsup|s+\<delta\>>>.
    </equation*>

    Furthermore, if we denote by <math|W<rsup|s,p>,s\<in\>\<bbb-R\>,p\<in\><around*|[|1,\<infty\>|]>>
    the fractional Sobolev spaces defined by the norm
    <math|<around*|\<\|\|\>|f|\<\|\|\>><rsub|W<rsup|s,q>>\<assign\><around*|\<\|\|\>|<around*|\<langle\>|\<mathD\>|\<rangle\>><rsup|s>f|\<\|\|\>><rsub|L<rsup|q>>>,
    then, for any <math|q\<in\><around*|[|1,\<infty\>|]>>,

    <\equation*>
      <around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,q><rsup|s>>\<leqslant\><around*|\<\|\|\>|f|\<\|\|\>><rsub|W<rsup|s+\<delta\>,p>>\<leqslant\><around*|\<\|\|\>|f|\<\|\|\>><rsub|B<rsub|p,\<infty\>><rsup|s+2\<delta\>>>.
    </equation*>
  </proposition>

  <\bibliography|bib|tm-abbrv|qft-drift>
    <\bib-list|23>
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    </bib-list>
  </bibliography>

  \;

  \;
</body>

<\initial>
  <\collection>
    <associate|info-flag|detailed>
  </collection>
</initial>

<\references>
  <\collection>
    <associate|FractionalLeibniz|<tuple|16|18>>
    <associate|L4estimate|<tuple|17|18>>
    <associate|Lemma:changeofvariables|<tuple|4|5>>
    <associate|Qcubeestimate|<tuple|14|16>>
    <associate|adjointparaproduct|<tuple|29|21>>
    <associate|analyticestimate1|<tuple|18|18>>
    <associate|analyticestimate2|<tuple|19|19>>
    <associate|analyticestimate3|<tuple|20|20>>
    <associate|analyticestimate4|<tuple|21|20>>
    <associate|analyticestimate5|<tuple|22|20>>
    <associate|auto-1|<tuple|1|1>>
    <associate|auto-10|<tuple|34|22>>
    <associate|auto-2|<tuple|2|2>>
    <associate|auto-3|<tuple|1|3>>
    <associate|auto-4|<tuple|3|4>>
    <associate|auto-5|<tuple|3.1|7>>
    <associate|auto-6|<tuple|3.2|7>>
    <associate|auto-7|<tuple|4|12>>
    <associate|auto-8|<tuple|5|18>>
    <associate|auto-9|<tuple|A|20>>
    <associate|bcross1|<tuple|20|16>>
    <associate|besovembedding|<tuple|34|22>>
    <associate|bib-MWcomedown|<tuple|21|23>>
    <associate|bib-albeverio_invariant_2017|<tuple|2|22>>
    <associate|bib-albeverio_remark_2008|<tuple|1|?>>
    <associate|bib-bahouri_fourier_2011|<tuple|3|22>>
    <associate|bib-barashkov_gubinelli_variational|<tuple|4|22>>
    <associate|bib-benfatto_ultraviolet_1980|<tuple|5|22>>
    <associate|bib-brydges_new_1983|<tuple|6|22>>
    <associate|bib-catellier_paracontrolled_2013|<tuple|7|22>>
    <associate|bib-feldman_lambda_1974|<tuple|8|22>>
    <associate|bib-feldman_wightman_1976|<tuple|9|22>>
    <associate|bib-glimm_positivity_1973|<tuple|10|22>>
    <associate|bib-glimm_quantum_1987|<tuple|11|22>>
    <associate|bib-gubinelli_global_2019|<tuple|13|22>>
    <associate|bib-gubinelli_paracontrolled_2015|<tuple|14|22>>
    <associate|bib-gubinelli_pde_2018|<tuple|12|22>>
    <associate|bib-gulisashvili_exact_1996|<tuple|15|23>>
    <associate|bib-hairer_theory_2014|<tuple|16|23>>
    <associate|bib-jona_lasinio_stochastic_1985|<tuple|17|23>>
    <associate|bib-kupiainen_renormalization_2016|<tuple|18|23>>
    <associate|bib-magnen_infinite_1976|<tuple|19|23>>
    <associate|bib-moinat_space_time_2018|<tuple|20|23>>
    <associate|bib-parisi_perturbation_1981|<tuple|22|23>>
    <associate|bib-park_lambda_1975|<tuple|23|23>>
    <associate|boundIu|<tuple|16|5>>
    <associate|commutatorestimate|<tuple|28|21>>
    <associate|compactembedding|<tuple|25|21>>
    <associate|eq:Phi4def|<tuple|17|7>>
    <associate|eq:bd|<tuple|9|?>>
    <associate|eq:contr-prob|<tuple|8|?>>
    <associate|eq:def-V|<tuple|7|3>>
    <associate|eq:def-Y|<tuple|4|2>>
    <associate|eq:drift|<tuple|13|4>>
    <associate|eq:drift-uu|<tuple|14|5>>
    <associate|eq:flat|<tuple|6|?>>
    <associate|eq:girs|<tuple|2|?>>
    <associate|eq:map-U|<tuple|15|5>>
    <associate|eq:phi43|<tuple|1|1>>
    <associate|eq:pre-drift|<tuple|12|4>>
    <associate|eq:pre-pre-drift|<tuple|11|4>>
    <associate|eq:renorm-drift|<tuple|10|4>>
    <associate|eq:rw|<tuple|19|8>>
    <associate|eq:sde-girs|<tuple|3|?>>
    <associate|eq:th1|<tuple|1|3>>
    <associate|eq:uw|<tuple|18|8>>
    <associate|lemma:bounds2|<tuple|9|10>>
    <associate|lemma:driftgronwallestimate|<tuple|8|9>>
    <associate|lemma:estimateremainder|<tuple|10|11>>
    <associate|lemma:localestimate|<tuple|6|7>>
    <associate|multiplierestimate|<tuple|32|22>>
    <associate|paraproductestimate|<tuple|27|21>>
    <associate|paraproductleibniz|<tuple|33|22>>
    <associate|sec:analytic|<tuple|5|18>>
    <associate|sec:appendix-para|<tuple|A|20>>
    <associate|sec:lp-bounds|<tuple|3.2|?>>
    <associate|symbolinq|<tuple|21|22>>
    <associate|table:reg|<tuple|1|3>>
    <associate|theta|<tuple|5|3>>
    <associate|thm:equicoerc|<tuple|3|4>>
  </collection>
</references>

<\auxiliary>
  <\collection>
    <\associate|bib>
      albeverio_remark_2008

      glimm_positivity_1973

      feldman_lambda_1974

      park_lambda_1975

      feldman_wightman_1976

      magnen_infinite_1976

      benfatto_ultraviolet_1980

      brydges_new_1983

      parisi_perturbation_1981

      jona_lasinio_stochastic_1985

      hairer_theory_2014

      catellier_paracontrolled_2013

      kupiainen_renormalization_2016

      MWcomedown

      gubinelli_global_2019

      albeverio_invariant_2017

      moinat_space_time_2018

      gubinelli_pde_2018

      barashkov_gubinelli_variational

      barashkov_gubinelli_variational

      glimm_quantum_1987

      barashkov_gubinelli_variational

      barashkov_gubinelli_variational

      gulisashvili_exact_1996

      bahouri_fourier_2011

      gubinelli_paracontrolled_2015
    </associate>
    <\associate|table>
      <tuple|normal|<surround|<hidden-binding|<tuple>|1>||Regularities of the
      various stochastic objects, the domain of the time variable is
      understood to be <with|mode|<quote|math>|<around*|[|0,\<infty\>|]>>,
      <with|mode|<quote|math>|C<with|math-font|<quote|cal*>|C><rsup|\<alpha\>>=C<around*|(|<around*|[|0,\<infty\>|]>;<with|math-font|<quote|cal*>|C><rsup|\<alpha\>>|)>>
      and <with|mode|<quote|math>|L<rsup|2><with|math-font|<quote|cal*>|C><rsup|\<alpha\>>=L<rsup|2><around*|(|\<bbb-R\><rsub|+>;<with|math-font|<quote|cal*>|C><rsup|\<alpha\>>|)>>.
      Estimates in these norms holds a.s. and in
      <with|mode|<quote|math>|L<rsup|p><around*|(|\<bbb-P\>|)>> for all
      <with|mode|<quote|math>|p\<geqslant\>1> (see
      [<write|bib|barashkov_gubinelli_variational><reference|bib-barashkov_gubinelli_variational>]).>|<pageref|auto-3>>
    </associate>
    <\associate|toc>
      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Introduction>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-1><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>The
      setting> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-2><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Construction
      of the drift measure> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-4><vspace|0.5fn>

      <with|par-left|<quote|1tab>|3.1<space|2spc>Proof of absolute continuity
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-5>>

      <with|par-left|<quote|1tab>|3.2<space|2spc><with|mode|<quote|math>|L<rsup|p>>
      bounds <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-6>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>Singularity
      of <with|mode|<quote|math>|\<Phi\><rsub|3><rsup|4>> w.r.t. the free
      field> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-7><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|5<space|2spc>Some
      analytic estimates > <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-8><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Appendix
      A<space|2spc>Besov spaces and paraproducts>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-9><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Bibliography>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-10><vspace|0.5fn>
    </associate>
  </collection>
</auxiliary>