> <\body> <\hide-preamble> >||>|>>> > \ <\with|par-sep|0.2fn> for the exponential Euclidean QFT on the plane>|C.De Vecchi, M.Gubinelli, M.Turra>||De Vecchi>|<\author-affiliation> Department of Mathematics, University of Pavia Via Adolfo Ferrata 5, 27100 Pavia, Italy |<\author-affiliation> Institute for Applied Mathematics &\ Hausdorff Center for Mathematics, University of Bonn\ Endenicher Allee 60, 53115 Bonn, Germany |>>|||<\author-affiliation> Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG Oxford, United Kingdom \; |<\author-affiliation> Institute for Applied Mathematics &\ Hausdorff Center for Mathematics, University of Bonn\ Endenicher Allee 60, 53115 Bonn, Germany |>>||<\author-affiliation> Institute for Applied Mathematics &\ Hausdorff Center for Mathematics, University of Bonn\ Endenicher Allee 60, 53115 Bonn, Germany |>>> We give a novel characterization of the Euclidean quantum field theory with exponential interaction > on> through a renormalized integration by parts (IbP) formula, or otherwise said via an Euclidean Dyson\USchwinger equation for expected values of observables. In order to obtain the well-posedness of the singular IbP problem, we import some ideas used to analyse singular SPDEs and we require the measure to \Plook like\Q the Gaussian free field(GFF) in the sense that a suitable Wasserstein distance from theGFF is finite. This guarantees the existence of a nice coupling with the GFF which allows to control the renormalized IbP formula. ||> <\small> <\table-of-contents|toc> Introduction> |.>>>>|> |.>>>>|> > |.>>>>|> > |.>>>>|> > Formulation of the problem> |.>>>>|> Reformulation of Problem |-0.3em|>|0em||0em|>> |.>>>>|> > A priori deductions for exponential interaction |.>>>>|> > A description of the exponential measure |.>>>>|> > Uniqueness of solution> |.>>>>|> Proof of uniqueness of solutions to Problem |-0.3em|>|0em||0em|>> |.>>>>|> > Analysis of the resolvent equation |.>>>>|> > Proof of uniqueness of solutions to Problem |-0.3em|>|0em||0em|>> |.>>>>|> > Existence of solutions via Lyapunov functions> |.>>>>|> Lyapunov functions |.>>>>|> > Measure of the approximating problem |.>>>>|> > Tightness of the measure |.>>>>|> > Limit of the operator |.>>>>|> > Limit as +\> |.>>>>|> > Limit as +\> |.>>>>|> > Limit as \0> |.>>>>|> > Besov spaces and heat semigroup> |.>>>>|> Stochastic estimates for the Wick exponential> |.>>>>|> Estimates on linearized PDEs> |.>>>>|> Proof of Lemma |-0.3em|>|0em||0em|>>> |.>>>>|> Technical results for the resolvent equation> |.>>>>|> Flow equations |.>>>>|> > Derivatives of the flow |.>>>>|> > Existence and equations |.>>>>|> > Properties of the flow derivatives |.>>>>|> > |.>>>>|> \; One of the first steps in constructive quantum field theory (QFT) is to build a family of distributions satisfying Wightman axioms, which can thus be interpreted as the Wightman functions of a field theory with unique ground state, invariant with respect to the Poincaré group. Wightman axioms, however, do not identify uniquely a particular QFT. Instead different QFTs are expected to satisfy different : a systems of partial differential equations (PDEs) relating Wightman functions and encoding the local and hyperbolic equations of motions for the quantum fields. An important progress in the analysis of Wightman QFTs was the introduction of Schwinger functions, namely the analytic continuations on imaginary time of Wightman functions, which are described by a set of axioms introduced by Osterwalder and Schrader (see). As observed by Nelson (see), in many cases (such as the scalar bosonic QFT) Schwinger functions are the moments of a probability measure > on Schwarz distributions (see for systematic applications of this approach). In particular, the Dyson\USchwinger equations translate in an integration by parts (IbP) formula for the measure > and it becomes natural to investigate the problem of existence and uniqueness of probability measures satisfying prescribed integration by parts formulas. The characterization of a measure through some IbP formula is a classical subject in stochastic analysis which has different formulations, such as existence and uniqueness of a measure with given logarithmic gradient or the unique closability of a minimally defined pre-Dirichlet form (and the references therein). The application of logarithmic gradients and integration by parts formulas to quantum field theory was already proposed by Kirillov in the case of sine\UGordon models, see where the problem is considered without renormalization. In the aforementioned works by Kirillov, the author exploits also a Lyapunov functions technique to show existence of solutions to the integration by parts formulas; a generalization of this technique is applied to the related problem of non-singular (i.e.with no need for renormalization) Fokker\UPlanck\UKolmogorov equations by Bogachev and Röckner. The problem of uniqueness of solutions to integration by parts formula or, similarly, to an infinite-dimensional Fokker\UPlanck\UKolmogorov equation, is solved only in some particular cases, see for instance the books, and the works by Bogachev, Da Prato and Röckner and by Röckner, Zhu and Zhu, where a dissipative non-regular drift (without renormalization) is considered. Let us also mention the studies about uniqueness of solutions to Fokker\UPlanck\UKolmogorov equations or of invariant measures of the |)>> stochastic quantization equation on the two-dimensional torus. Although in this case the problem of existence and uniqueness \ is solved, it is not clear whether the techniques used in the aforementioned papers can be extended to the models on the non-compact space > or in dimension greater thantwo. It is worth to mention that the study of uniqueness in the framework of Dirichlet forms for the exponential model has been discussed by Albeverio, Kawabi and Röckner in for the one-dimensional non-singular case, and by the same authors together with Mihalache in for the two-dimensional setting on the torus. See also for a review of the existing literature on the Dirichlet form approach to the dynamical problem. Let us mention that our concern here is different from that in Dirichlet form theory since here the measure is an unknown in the problem and not given a priori. The key open problem that we address in this paper is to provide a suitable setting in which existence and uniqueness of measures satisfying given some IbP formula, that is one involving a renormalization procedure in its definition, which is the usual situation in constructive Euclidean QFT. Instead of attempting a general framework, we concentrate in a particular case where we can establish a reasonable well-posedness theory for the singular IbP problem: we test our ideas on the EQFT with exponential interaction and positive mass 0>, or Høegh\UKrohn model on the whole space >. The exponential interaction in the case of mass is a classical example of conformal field theory and it finds important applications in Liouville quantum gravity. As far as we know, this contribution of ours is the first which manages to address this question for an EQFT requiring renormalization and in the infinite volume limit.\ Let us give a more detailed description of the problem that we consider here. Let |)>> be the space of Schwartz functions and denote its dual, that is the space of tempered distributions, by|)>>. We fix a Banach space (or a topological vector space) \|)>> and we consider a family > of functions \> which are Fréchet differentiable. In particular, for any \E>, we can consider the derivative F|)>> of in the direction \|)>>; the map DF|)>> is linear and bounded in , and thus in |)>>. This means that, since |)>> is the topological dual of |)>>, there exists a unique >F|)>\\|)>> such that F|)>=|\>F|)>,f|\>>, where |\,\|\>=|\,\|\>,\>> is the duality between |)>> and |)>>.\ Let us denote by > a family of probability measures on , and let \|)>> a given map. Then a generic IbP problem, or Euclidean Schwinger\UDyson equation for a measure \\>, has the general form <\equation*> |\>F-F B,f|\>d\=0,F\Cyl. We are interested in functionals which have the form <\equation*> B|)>=p|)>,x\\, for some smooth function \\>. This locality of the IbP formula is peculiar of EQFT where locality (or reflection positivity, or domain Markov property) is structurally linked to the finite speed of propagation of signals in the Minkowski theory. Unfortunately, such kind of functions are seldom well-defined on the set on which we could hope that any solution > would be supported. Typically, this support looks very much like the support of the GFF and therefore non-linear local functionals are not automatically well-defined and need to be approached via an ultraviolet regularization and subsequent renormalization. In this sense, we talk about a IbP formula adopting the term from the recent literature in singular stochastic PDEs. Given this motivation we consider a sequence of maps >|)>\0>> such that, for every \0>, we have >:E\\|)>> and for which we recover locality in the limit as \0>. They are typically of the form <\equation*> B|)>=p>>\\|)>|)>,x\\ where >|)>\0>> is some sequence of local smoothing kernels for which >\\|)>\\> in |)>> and >:\\\|)>>> is a sequence of smooth function chosen to deliver the expected renormalization, typical of EQFT in two and three dimensions. The main problem we analyze in this paper can therefore be formulated abstractly as follows: <\problem|A> We say that a measure \\> satisfies the >|)>\0>> and>> if, for any \|)>>, we have <\equation> |\>F|)>,f|\>\\\|)>=lim\0>F|)>|B>|)>,f|\>\\|)>,F\Cyl, where > is the set of smooth and bounded cylinder functions (cf.the Notation section at the end of the present introduction). Let us remark that the problem of existence and uniqueness of solutions to Problem strongly depends on the subset > of the space > of (Radon) probability measures on. One of the main problems is that, if we consider =\>, i.e.we consider a generic Radon probability measure on , then it is not clear if >|)>>> admits a limit in probability as \0> and whether the limit depends on the measure > or not. For example, if we take >> to be the drift of > measure in >, namely <\equation*> B>|)>=>\\|)>-6c> >\\|)>+3c> with >=|+m|)>g>|\<\|\|\>>|)>>>, then it is known that >> converges to the unique limit \:>(where \:> is the Wick product of Gaussian random fields, see Chapter1 in) when the measure > is absolutely continuous with respect to the Gaussian free field with mass0>. On the other hand, if > is supported on the space of smooth functions, such a limit does not exist. It is useful then to consider a class of measures > for which it is possible to make sense (almost surely) of the limit \0>B>>. The set > can therefore be neither too large nor too small and we find useful to focus on measures which \Plook like\Q the Gaussian free field with mass 0>. We formalize this idea introducing an appropriate Wasserstein metric which encodes this proximity without requiring absolute continuity, which in the full |)>> is anyway never an appropriate condition. Let be a convex cone in equipped with a norm |\|\<\|\|\>>> that is stronger than the one of , then we define the function :\\\\|]>>as <\equation*> d\ \|)>=inf\\,\|)>>E>|x-y|\<\|\|\>> \ \x,\y|)>,\,\\, where ,\|)>\\|)>> is the set of couplings between > and >, and <\equation*> \\|x\B>>|>|x\B>>>>>. Let us suppose that contains the support of the measure > of the Gaussian free field with mass 0> and define <\equation> \\\\\d,\|)>\+\|}>. The space > strongly depends on the convex cone together with its norm. For example, if we consider |)>> we have\ <\equation*> \|)>>=\\\\\|}>. Another simple case is when equipped with its natural norm, this gives <\equation*> \=\, namelythe Wasserstein space of measures on (see Chapter6 in). The class of measures > encodes the existence of sufficiently regular couplings between our target measures and the GFF. This mirrors the situation in singular SPDEs and in other recent development in EQFT where the existence of such couplings has been a key technical aspect to develop a suitable stochastic analysis of singular dynamics and EQFTs. In the fundamental work, Da Prato and Debussche indeed introduced a notion of solution to stochastic quantization equations in two dimensions as the sum of the Gaussian free field with mass and a solution to a (random) non-linear PDE. In other words, they look at the solution as a perturbation of the solution to the linear stochastic heat equation. More recently, Barashkov and Gubinelli developed a coupling approach based on an optimal control problem, Bauerschmidt and Hofstetter used a similar coupling to study the pathwise properties of the two dimensional sine\UGordon EQFT on a torus and Shenfeld and Mikulincer linked these ideas with current developments in the theory of optimal transport and functional inequalities. Similar ideas are fundamental also in the context of singular stochastic PDEs, see regularity structures, paracontrolled calculus, and renormalization group theory. As we already mentioned, we would like to provide complete well-posedness results within this framework of singular IbP formulas and, for this reason, we focus on the specific case of the exponential interaction, namely we take +B>, where =C>>|)>>, i.e.the (weighted) Besov\UHölder space with negative regularity> (see Appendix), and > is taken to be <\equation*> B\B|)>-\>>|)>, where |)>\0> satisfies some conditions depending on the parameter> (see Definition below) and |)>-\>>> is a weighted Besov space (see Appendix). \ Moreover, we set <\equation> B>|)>\+m|)>\+\ f> e>\\|)>-|2> c>>, where ,m\\>, >> is a smooth, spatial cut-off function such that >\1>, >=\g\|)>> is a regular mollifier, and <\equation> c>\>g>+m|)>g> \z is a renormalization constant diverging logarithmically to> as \0>. Finally, we consider the space of measures > in Problem to be >> (see equation), that is an intermediate regime between the case >> (i.e.the space of measures that are absolutely continuous with respect to the Gaussian free field) and > (which coincides with the Wasserstein space>). <\remark> In the following, we shall consider the number \0.55> given by the maximum taken over all 1>, satisfying Definition,of <\equation> |r+1|)>>. Moreover, we let |~>\3-2\0.172\\>. In the present setting, it is possible to obtain the following results.\ <\theorem> Suppose that \4\|~>>. Consider >> (see Definition) with +B>, where <\equation*> B=C>>|)>B=B>\B|)>-\>>|)>\\\,f\0|}>. Then there exists a unique solution to Problem with respect to >|)>>> (given by equation) and the space of measures>>. It is also possible to obtain an existence result for the whole regime \8\>. <\theorem> Suppose that \8\> and assume that the same hypotheses on the spaces >, >, , >> and the drift >|)>>> as in Theorem hold. Then there exists a solution to Problem with respect to >|)>>> and>>. Theorem is obtained by building suitable Lyapunov functions independent of\0>, similarly to what was done by Kirillov for the not-renormalized equation (see), and reducing the infinite-dimensional problem to the existence of a symmetric invariant measure for a finite-dimensional differential operator.\ An important consequence of Theorem and Theorem is a differential characterization of the exponential measure (see Theorem for details). Let us mention that in the present paper we also get a uniqueness result for a slightly more restrictive formulation of Problem (cf.Problem in Section) in the regime \4\\>. The possible measures solving this latter formulation of the IbP problem contain the invariant measure of the stochastic quantization equation with exponential interaction. Let us present here the structure of the paper. In Section, we discuss the IbP Problem in the general setting, establishing some equivalent formulations that will be useful to address Problem itself. We also consider the more restrictive Problem and Problem, their relation with Problem and some properties of the solutions to such problems, such as the negativity of the coupling between the Gaussian free field and the quantum field measure with exponential interaction. Section is devoted to the study of uniqueness of solutions to Problem showing the uniqueness result stated in Theorem. Since the proof relies on some properties of the solution to the resolvent equation associated to the drift >>, most parts of the section is dedicated to the study of such an object. The existence of solutions to Problem is proved in Section. The proof is based on an approximation method and it involves Lyapunov functions. In Appendices and we recall the definitions and properties of weighted Besov spaces and Wick exponential, respectively, which are used throughout the paper. Appendices, , and are concerned with some technical, analytical results on (S)PDEs exploited in the paper. We fix here some notation adopted throughout the paper.\ We write b> or a> if there exists a constant 0>, independent of the variables under consideration, such that C b>, and b> if both b> and a>. If the aforementioned constant depends on a variable, say ), then we use the notation b>, and similarly for\. The space > is the space of linear and bounded functionals from the Banach space into the Banach space . > is equipped with its natural operator norm|\|\<\|\|\>>>>. We also write =L>.\ The space of Schwartz functions on > is denoted by |)>> and its dual, that is the space of tempered distributions, is denoted by|)>>. Moreover, we let |)>> be the space of -times differentiable functions on > with continuous and bounded derivatives. We also use the notation >|)>> to denote the weighted Besov space on > (see Appendix for more results on Besov spaces). For \0>, we define the weight >=|)>/2>>, \>, and >\\>>. Let be a convex subset of |)>>, we define the set > of on to be the set of functions \> such that there exist a function \C|)>>, with at most linear growth at infinity and all bounded derivatives, and ,\,u\\|)>>, such that |)>=|u,\|\>,\,|u,\|\>|)>>, for every \K>. We say that Cyl> has compact support in Fourier variables if the Fourier transforms > of >, ,n>, have compact supports on >. Moreover, we denote by > the subset of > such that, sticking with the previous notation, the function > is also bounded. We adopt the notation =e+m|)>t>> for the heat kernel with mass (cf.Appendix). The authors thank Sergio Albeverio for the kind suggestions and remarks on various early drafts of the paper. This work has been partially funded by the German Research Foundation (DFG) under Germany's Excellence Strategy - GZ 2047/1, project-id 390685813. The first author is partially funded by the Istituto Nazionale di Alta Matematica \PFrancesco Severi\Q (INdAM), Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA): \PAnalisi armonica e stocastica in problemi di quantizzazione e integrazione funzionale\Q. > In this section, we want to reformulate Problem when the space of measures > is the space > introduced in. For this reason, we consider +B>, where > is some space supporting the measure of the Gaussian free field with mass 0>, and =B>. We consider also the natural projection :B\B\B> and the map :B\B\E> such that \X+Y>. Let us reformulate Problem by means of a second order operator. Consider >> to be regular enough and let\ <\equation*> \>F\ tr|)>>>F|)>-|B>,\>F|\>,F\Cyl. <\problem|A'> We say that a measure \\> satisfies the >>>if <\equation> lim\0>|>F|)> G-F >G|)>|>\\=0,F\Cyl,G\Cyl. <\remark> A consequence of equation is that <\equation> lim\0>\>F\\=0,F\Cyl. That is what is usually called . We now want to lift the problem from the space to the space \B>. In order to do so, we introduce the following notion: <\definition> The subset of measures > satisfies the if, for any \\>, there exists a Radon measure > on \B> with the following properties: <\enumerate-roman> >\=\>, where > is the law of the Gaussian free field on >, >\=\>, |Y|\<\|\|\>>> \X,\Y|)>\+\>, we call > a > with the free field>. We denote by \B>> the set of Radon measures on \B> satisfying conditioni.andiii. <\remark> With the same notation as in Definition, it is clear that >> defined as in where >, coincides with the space>\\B>>. \ Indeed, \\>> if and only if there exists a coupling x,\z|)>> between z|)>> and the free field x|)>> such that the difference is supported on>. The coupling > is related with a measure > in \B>> via the transformation . We consider now an operator >> on the space of regular functions on \B> of the form <\equation*> \>\\ tr\|)>-|+m|)>X,\\|\>-|B>-+m|)>Y,\\|\>. <\problem|A''> We say that a measure \\>> satisfies the >>>if <\equation> lim\0>|\>P|)> G\P-F\P \>P|)>|>\\X,\Y|)>=0,F\Cyl,G\Cyl, where > is a coupling of > with the free field. <\remark> As in the case of Problem (see also Remark), equation implies <\equation> lim\0>\>P|)>\\X,\Y|)>=0,F\Cyl, We often say that > solves Problem if there is a measure > solving the FPK equation with > as coupling of > with the free field.\ Problem is based on a formulation of integration by parts formula related to stationary solutions to the Fokker\UPlanck\UKolmogorov equation, making the operator >> symmetric on its domain (see Problem in Section or for more details on the relations between the formulations). It is convenient to introduce new equations related to Problem described above, where the argument of >> in equation is not necessarily of the form P>. <\problem|B> We say that a Radon measure > on \B> satisfies the >>>, if <\equation> lim\0>\>\ \\=0,\\Cyl\B>. <\problem|Bsym> We say that > satisfies the >>> if, furthermore, <\equation> lim\0>|\>P|)> G\P-F\P \>P|)>|>\\\=0, for any +B\\> such that P\Cyl,G\P\Cyl>. We have then the following result. <\theorem> Suppose that \0>|B>|)>,\>F|\>|\|>\\\|)>\+\>, for any Cyl> and \\>. Then the following statements hold: <\enumerate-roman> Problem is equivalent to Problem. Assume further that =\>>, then Problem, Problem and Problem are equivalent. Assume the same hypothesis as in pointii.. Then, a solution to Problem is a solution to Problem. Summarizing, if all the hypotheses of pointsi., ii., andiii., then Problem is equivalent to Problem, and Problem implies Problems and, thatis <\equation*> >\>\>\> <\proof> Let us give the proof point by point. > Let us prove that a solution to Problem is also a solution to Problem. Let \\> solve Problem, take Cyl>, and consider |)>=|\,f|\>>, for \|)>>. Then, by equation>, we have <\equation*> lim\0>\>G|)>\\=0, where\ <\eqnarray*> >G|)>>||> G+F \>+2|\>F,\>G|\>>>|||> G+F \>+2|\>F,f|\>.>>>> Thus, we have <\eqnarray*> ||\0>\>G|)>\\=lim\0>> G+F \>+2|\>F,f|\>|)>\\>>|||\0>>+2|\>F,f|\>|)>\\=lim\0>| B>,f|\>+2|\>F,f|\>|)>\\.>>>> We now show that any solution to Problem is a solution to Problem. Let \\> solve Problem. Consider an orthonormal basis |)>> of |)>> and take |\G,e|\>>, for some Cyl>. We have then, by equation, <\equation*> \G,e|)>\\=|\F,e|\>\\=|\G,e|\>|B>,e|\>\\. Taking the sum over \> on both sides, noticing that by the properties of cylinder functions we can exchange the integral with the sum, and exploiting Parseval's identity we have <\equation*> tr|)>>\G|)>\\=|\G,B>|\>\\, that is \>G \\=0>. Now, consider again an orthonormal basis |)>> of |)>>, but take |\H,e|\>>, for some Cyl>. Let us note that <\equation*> |\F,e|\>=|\|\H,e|\>|)>,e|\>=|\G,e|\>|\H,e|\>+G \H,e|)>. On the other hand, equation gives <\equation*> |\|\H,e|\>|)>,e|\>\\=G|\H,e|\>|B>,e|\>\\. This yields <\equation*> G|\H,e|\>|B>,e|\>\\=||\G,e|\>|\H,e|\>+G \H,e|)>|>\\, and taking the sum over \> and using again the properties of cylinder functions, we get <\equation*> G|\H,B>|\>\\=||\G,\H|\>+G tr|)>>H|)>|>\\, which implies <\equation*> 0=||\G,\H|\>+G trH|)>-G|\H,B>|\>|>\\=||\G,\H|\>+G \>H|>\\, namely\ <\equation*> G \>\\=-|\G,\H|\>\\. Doing the same computation exchanging the roles of and we also get H \>\\|)>=-|\G,\H|\>\\|)>>, that gives <\equation*> G \>\\-H \>\\=0, and concludes the proof of the first point of the theorem. > Assume further that =\>>. Notice that >P|)>=\>\P>, and since >\=\> we have that <\equation*> \>P|)>\\=\>\P>\\=\>F\\, which gives the equivalence between equations and and in the same way we have the equivalence between equations and. > Let > be a solution to Problem. We only have to show that \\\B>>, that is, the only condition that we need to verify is that >\=\>. Since the terms involving derivatives with respect to of >> do not depend on >, in this proof we write > for the operator>>. Furthermore, for a solution > to\U we have that, if > does not depend on, <\equation*> 0=lim\0>\>\ \X,\Y|)>=\>\ \X,\Y|)>=\\ \X,\Y|)>. We start with the proof that the free field measure with mass 0> solves equations\U. Consider, for \B>,\ <\equation*> \\X=PX+P\ \s, and let >> be the limit\ <\equation*> \>X|)>=lim+\> \X|)> \t. We want to show that \\ \\>=0>, for any > smooth cylindric function. By Itô's formula (see Theorem4.32 in), taking expectation and dividing by 0>, we have that <\eqnarray*> \|)>-\|)>|]>>||\\|)>|]>\\s>>|||\\ \X|)> \\s>>|||\\ \|)> \\s|)>X|)>,>>>> and letting +\>, recalling that > is bounded, we get <\equation*> 0=\\ +\> \|)> \\s|)>X|)>=\\ \>X|)>. We now show uniqueness of the measure>. Consider <\equation*> G>|)>=\>>es>F|)> \s, where Cyl>> of the form <\equation*> F=|u,X|\>,\,|u,X|\>|)>. Let us prove that >G>> is well-defined, recall <\equation*> \>G>|)>=\G>|)>= tr>G>|)>-|+m|)>X,\>G>|\>, where\ <\equation*> \>G>=\>>es>|\>F|)>,\>X|\> \s=\>>es>\|u,X|\>,\,|u,X|\>|)> Pu \s, and\ <\eqnarray*> >G>>||>es>\>|\>F|)>,\>X|\>|]> \s>>|||>>es>\|u,X|\>,\,|u,X|\>|)> |Pu,Pu|\> \s,>>>> since <\equation*> \>X=Ph,\>X=0. This gives <\eqnarray*> G>|)>>|| tr|)>>|\>>es>\|u,X|\>,\,|u,X|\>|)> |Pu,Pu|\> \s|>>>|||>>es>\|u,X|\>,\,|u,X|\>|)>|+m|)>X, Pu|\> \ \s,>>>> and thus G>> is well-defined. 2.b.> We now show that <\equation> -\|)>G>\\\=F\\\. By Itô formula (see Theorem4.32 in), we have <\equation*> et>G>|)>-G>|)>=es>\>G>|)> \s+es>\>G>|)> \X-\es>G>|)> \s, and taking the expectation we get\ <\equation> et>\>|)>|]>-\>|)>|]>=\es>\>G>|)> \s-\\es>G>|)> \s. On the other hand,\ <\equation*> et>G>|)>-G>|)>=>e>\>|)>|]> \s->es>\>|)>|]> \s, and taking the expectation <\equation> et>\>|)>|]>-\>|)>|]>=-\es>\>|)>|]> \s. Comparing equations and, we get <\equation*> \\es>G>|)> \s-\es>\>G>|)> \s=\es>\>|)>|]> \s, dividing by 0> and letting 0>, we have the result.\ Take \> and \\>>, and call > and >> the solutions to equation <\equation*> -\+m|)>\=\, with initial conditions =Z> and =Z>, respectively. We show that \\>>. Consider the product measure \\>> between > and>>. Since we are working in separable spaces, any Borel measure is a Radon measure (see, e.g.Theorem9 in Chapter2, Section3 in), and therefore we have that, for every \0>, there exists 0> such that <\equation> \\\>|Z|\<\|\|\>>\r,|Z|\<\|\|\>>\r|)>\\. Since <\eqnarray*> |u,\|\>,\,|u,\|\>|)>-|u,\>|\>,\,|u,\>|\>|)>|\|>>|>||\|\<\|\|\>>>>|u,\-\>|\>|\|>>>||>||\|\<\|\|\>>>>|u|\<\|\|\>> |P|)>|\<\|\|\>>>>||>|>et>et> |Z|\<\|\|\>>+|Z|\<\|\|\>>|)> \t>>||>||\+m> |Z|\<\|\|\>>+|Z|\<\|\|\>>|)>,>>>> then we have, recalling \G> \\>=F \\>>, <\equation*> \G> \\-F \\>=\G>-\G>|)>|]>\C|\+m> \|Z|\<\|\|\>>+|Z|\<\|\|\>>|)>. The right-hand side of the previous inequality converges to zero as \+\> by property, and thus F \\>=F \\> since, by equation and being > a solution to Problem, we have G> \\=F \\>. This yields \\>> and uniqueness of the measure. Let ,m\\>, and \\/|)>>. <\definition> We define <\equation*> B\C>>|)>, for some \0> small enough, and with \0>. Let us recall that > denotes the presence of a weight, in fact >>|)>> and >>>|)>> are weighted Besov spaces (see Appendix). Also, we let > to be a space of Besov functions with positive regularity, in particular, we choose <\equation*> B\B>>|)>, where p\+\> and 0> satisfies the following condition depending on>: <\equation*> 0\s\\+2->, arising when dealing with the problem of existence of solutions to the Fokker\UPlanck\UKolmogorov equation (cf.Theorem). We suppose further that there exists 1> such that\ <\equation*> +\1,s-\-2\\0,\r\2. Notice that such a condition is always satisfied for some 0>, 1> and 1>, whenever \2>. Let us also recall the definition of the space\ <\equation> B>\B>>|)>\\\,f\0|}> featured in Theorem. Let us introduce the functional >:B\B\\|)>> <\equation> \>\\ f> \e>\X|)>>:e>\Y|)>>. where the term e>\X|)>>:> is defined as <\equation> \e>\X|)>>:=e>\X|)>-|2> c>>, with the constant >> introduced in equation. Here, >> is a smooth cut-off function with compact support and with derivatives uniformly bounded in > such that >\1> uniformly with respect to a polynomially weighted norm, and we let >=\g\|)>>, where is a positive, smooth, compactly supported function on > with Lebesgue integral equal to, and such that there exists its convolutional square root >>, i.e.>=>\>>, which is also positive, smooth, and compactly supported function with > \x=1>. We also assume that >=g>>, for all \>. Notice that this last property implies that >> takes real values. Consider an operator >> of the form <\equation> \>\= tr\|)>-|+m|)>X,\\|\>-|+m|)>Y+\>,\\|\>, which is well-defined for :B\B\\> in a suitable class > of regular functions to be specified below (see Definition). For the moment, we only require the set \B>> of smooth cylindrical functions to be contained in the family>. <\remark> Our choice of > and > implies that the space > contain the Gaussian free field with mass and that <\eqnarray*> +m|)>X>|>|>>|)>,>>|+m|)>Y>|>|>>|)>,>>|expX|)>:>|>|-\>>|)>,r\\2,>>|Y|)>>|>|>|)>\B,Y\0,>>>> where all the parameters are as in Definition (see and Appendix). For simplicity, we adopt the notation <\equation*> B,|~>>>=B-1|)>-\>,,|~>>|)>, and we omit the parameters whenever =r> and |~>=\>, i.e.>=B,|~>>>>. Let us introduce also the notation <\equation*> \=\\\ \eX>:eY>, where the measurable function expX|)>:> is defined on a subset of > of full measure (with respect to the free field measure with mass ) as the limit of expg>\X|)>:> as \0> (see Proposition for details). The operator >> defined in equation is an approximation for the operator\ <\equation> \\\ tr|)>> \|)>-|+m|)>X,\\|\>-|+m|)>Y+\,\\|\>, for \Cyl\B>>.\ <\theorem> Let > and > be as in Definition with the additional condition =B>> and consider a Radon measure \\\B>>. Then, for any ,\\Cyl\B>>, we have\ <\equation> lim\0>>\|)>\\\\=\|)> \\\\. <\proof> We show the proof only for the case =1>, the general case can be deduced via Lebesgue's dominated convergence theorem. Let us prove that \\L|)>>, for \Cyl\B>>. By definition, we have that |)>> \|)>> is bounded, and therefore in |)>>. Moreover,\ <\equation*> |+m|)>X,\\|\>|\|>=|X,+m|)>\\|\>|\|>\|X|\<\|\|\>>>|\\|\<\|\|\>>>>>, and since |X|\<\|\|\>>>\L|)>>, >\> being the Gaussian free field, and |\\|\<\|\|\>>>>>\L>\B,\|)>>, then also the term |+m|)>X,\\|\>> is in|)>>. Now, the term |+m|)>Y,\\|\>> can be handled similarly by using the hypothesis |Y|\<\|\|\>>>\L|)>>. We are left to consider the term |\,\\|\>=|\\eX>:eY>,\\|\>>. First of all, we note that the product expX|)>:expY|)>> is well-defined. Indeed, since >\> is the Gaussian free field we can exploit Proposition to get expX|)>:\B>> (cf.Remark). Furthermore, since 0> and B>>|)>>, by composition of a smooth bounded function with a Besov function we have Y|)>\B>>|)>>. Thus, by Theorem, provided <\equation*> >\+\1,s-|4\>-2\\0, we have that expX|)>:expY|)>\B,\+\>>, and the product is well-defined and continuous. On the other hand, since we have also Y|)>\L>|)>>, then Proposition implies that expX|)>:expY|)>\B>> and\ <\equation*> |\eX>:eY>|\<\|\|\>>>>\|\eX>:|\<\|\|\>>>>. Thus,\ <\equation*> |\,\\|\>|\|>\|\eX>:|\<\|\|\>>>> |\\|\<\|\|\>>>|)>>> . By hypothesis we have |\\|\<\|\|\>>>|)>>>\L>\B,\|)>> and, since \8\> by our assumptions in Definition, we also have that |\expX|)>:|\<\|\|\>>>>\L|)>>. Hence, <\equation*> |\,\\|\>|\|>\L|)>. We are left to show that equation holds. This is equivalent to showing <\equation*> lim\0>>\-\\|)>\ \\=lim\0>|\>-\,\\|\>\ \\=0. By the same reasoning as in the previous part of the proof and by Proposition, we have that |\>-\,\\|\>\L|)>>, uniformly with respect to\\1>. In order to show that the previous limit holds, it is sufficient to prove that\ <\equation*> |\>-\,\\|\>\0, in probability as \0>. On the other hand, we know that\ <\equation*> |\>-\|\<\|\|\>>>>\+\, uniformly with respect to>, and by Proposition we have that \ >\\>, weakly in |)>>. Therefore, we have that >\\> weakly in >>, and in probability with respect to>. By definition of weak convergence and since \\>|)>>> by hypothesis (see Definition), we have that |\>-\,\\|\>\0>, in probability. Finally, by propertyin Definition and the fact that <\equation*> |\>-\|\<\|\|\>>>>\L|)> uniformly in \0> by Proposition, we have that |\>-\,\\|\>\L|)>> uniformly in \0>, for some 1>, and thus the result follows from Lebesgue's dominated convergence theorem. <\remark> By Theorem it is possible to take the limit \0> inside the integration with respect to > in equations\U and\U so that the mentioned equations are equivalent to the one where the limit disappears and the operator >> is replaced by>. Another consequence of the previous theorem is that, if we consider |~>>> to be another approximation of > such that\ <\equation> lim\0>|~>>\|)>\\\=\|)>\\\,\,\\Cyl\B>, then Problems and with the operator |~>>> are equivalent to Problems and with the operator >>. This means that the formulations given in Problems and do not depend on the precise form of the approximating operator >> but only on its limit>. We now want to prove a result that justifies our restriction in taking =B>> in the results above and in the rest of the paper. Indeed, if we focus on Problem, the solutions are always supported on the space of negative functions on the component. In the light of pointin Theorem, the next result implies that any solution > to Problem belongs to the spaces\B>>. <\theorem> Let > be a solution to Problem, then we have <\equation*> supp P>|)>\0|}>. <\proof> Suppose that B=B>>|)>> (see Definition). Define, for 0>, <\equation*> \=|)>,x\\, and <\equation*> f=x\0,I=f \y,x\\. We consider, for \0>, the convolution >\Y> which, by definition of >, belongs to>>> for any ,s\1> such that \2>. Take <\equation*> F=arctan |I>\Y|)>,\|\>|)>, so that, by integration by parts, <\eqnarray*> ||>F>|||I>\Y|)>,\|\>> |f>\Y|)>,\ g>\+m|)>Y-\>|)>|\>>>||||I>\Y|)>,\|\>> |f>\Y|)>,\+m|)>>\Y|)>+g>\\>|)>|\>>>||||I>\Y|)>,\|\>> ||f>\Y|)>,\ m>\Y|)>|\>+|f>\Y|)>,\|)>>\Y|)>|\>||1>>>>||||1>|+|f>\Y|)>,\>\\>|)>|\>|>>>||||I>\Y|)>,\|\>> ||f>\Y|)>,\ m>\Y|)>|\>+|f>\Y|)>>\\Y|)>,\>\\Y|)>|\>||1>>>>||||1>|+|f>\Y|)>,\\ >\\Y|)>|\>+|f>\Y|)>,\>\\>|)>|\>|>.>>>> Consider the term |f>\Y|)>,\\ >\\Y|)>|\>>, and multiply and divide by > to get\ <\equation*> |f>\Y|)>,\\ >\\Y|)>|\>=|\ f>\Y|)>,\|\> >\\Y|)>|\>. By Young inequality, we have\ <\eqnarray*> >F>|>||I>\Y|)>,\>|\>> ||f>\Y|)>,\ m>\Y|)>|\>+|f>\Y|)>>\\Y|)>,\>\\Y|)>|\>||2>>>>||||-|\ f>\Y|)>,|\|\>|> |\>-|\ , >\\Y|\|> \>\Y\0|}>>|\>|>,>>>> where we also used that the term |f>\Y|)>,\>\\>|)>|\>> is positive. We want to show that \/\> is bounded. Taking x>, we have, for 0> small enough, <\equation*> \|\>=|1+k>= y|1+>\2 sup\>|1+>\C, which gives, if /2-m\-\>, <\equation*> \>F\-|I>\Y|)>,\>|\>> |\|f>\Y|)>,\ |\>+|\, >\\Y|\|>\>\Y\0|}>>|\>|>, since >\Y|)>=g>\Y> on >. Since the right-hand side does not depend on>, and the left-hand side converges to when integrated with respect to>, we have\ <\equation*> |1+|I>\Y|)>,\>|\>> ||f>\Y|)>,\ |\>+|\, >\\Y|\|>\>\Y\0|}>>|\>|>\\\0, and since the terms |\, >\\Y|\|>\>\Y\0|}>>|\>>, >, >\\Y|\|>>, and |I>\Y|)>,\>|\>|)>> are all positive, we have\ <\equation*> \|f>\Y|)>,\ |\> \\\0. By Fatou's Lemma we can take the limit as \0> and get that <\equation*> \0|}>>=0,\ <\remark> A consequence of Theorem is that for the case of exponential interaction, a solution to Problem with =B>>|)>> is necessarily a solution to Problem with =B>>. What has been described in the previous section has the main aim of giving a characterization to a unique measure which we call the exponentialQFT. In this section, we want to connect our discussion of the IbP formula with the standard approach to construct EQFT via Gibbsian modifications of theGFF. Assume that > is the measure of the free field with mass on the torus > of size. We define the interaction, for \0>, <\equation*> V>>|)>\>f>e>\\|)>-|2>c>>\x, with , , >> and > are as in Section. We also consider <\equation*> Z>\e>>|)>> \\|)>. <\definition> We say that the measure >> on |)>> is the measure related with the Euclidean QFT having action <\equation*> S|)>= >\|\|>+m\|)> \x+>\e\>:\x, if there are two sequences \0> and >\+\>, with \0> and \\> such that <\equation> \>\|)>=lim\+\> lim+\> Z>,\> e>,\>>|)>>\>>\|)>, where the limits are taken in weak sense in the space of probability measures. Such a measure >> was first built by Albeverio and Høegh-Krohn in (see also) using techniques from constructive QFT. More recently, this model was studied in the context of stochastic quantization on the torus or on a compact manifold (see e.g.), and on> (see). See also for the related |)>>-model. A consequence of the results we presented in the previous sections of the paper (see Theorem, Theorem) is the following differential characterization of the measure related to the exponential interaction. <\theorem> The following statements hold: <\enumerate-roman> For any \8\>, there exists a measure related to the exponential interaction defined by the limit. Let =C>>|)>> and =B>>.Then, for any \4\|~>>, the measure >> is the unique measure in the space +B>> such that, for any Cyl> and \|)>>, we have <\equation*> |\>F,h|\>\\\|)>=lim\0>F|)> |+m|)>\+\f>eg>\\-|2>c>>,h|\>\\\|)>. > In this section, we discuss the uniqueness of solutions to Problems,, and in the case where >, >, =\>>, and +B> are as in Section and >> is the drift of the exponential interaction (see equation). The method that we adopt to prove uniqueness is the study of the resolvent equation for the operator >> defined in equation, namely\ <\equation> -\>|)>G>>=F,\\\,. We are interested in classical solutions to the previous equation, namely solutions >>> that are at least in \B|)>> and to which the operator >> can be applied. Let us first introduce here a class of functions where the solutions of the resolvent equation belongs. <\definition> Recall that =\/|)>>. Let ,r> be the same parameters as in Definition and let us introduce the space\ <\equation*> =B>,,\>|)>, for some \+\> large enough. Note that \>. Denote by > the class of bounded, measurable functions :\B\\> such that\ <\enumerate-roman> \\C\B,B\|)>+\>>|)>|)>>, for any \0>, l\+\>. \\C\B,B>>|)>|)>>, so that |\\,+m|)>X|\>> is well-defined. The operator \\C\B,L,B|)>|)>> can be extended in a unique continuous way to an operator \\C\B,L>>|)>,H>>|)>|)>|)>>, where \1> and \1>. There exists some >\L|)>>, |)>>, such that <\eqnarray*> |\\|\<\|\|\>>B\|)>+\>>|)>>>|>|>,>>||\\|\<\|\|\>>>>|)>>>|>|>,>>||\\|\<\|\|\>>>>|)>,H>>|)>|)>>>|>|>.>>>> We define the space > as the space of functions +B\\> such that <\equation*> F\P\\. <\remark> Under conditionof the previous definition, since the immersion >>|)>\L|)>> is an Hilbert-Schmidt operator, then \,\>\\\L>\B,TC|)>|)>>, where |)>> is the space of trace-class operators on|)>>. <\remark> The classes > and > are chosen in such a way that they satisfy the following two important properties: (i) > and > contain \B>> and >, respectively, and (ii) if =B>> then, for any Radon measure \\\B>>, we have \0>>\|\|>>\\\+\>, for any \\> and \1>. The classes > and > satisfy the following important lemma. <\lemma> Suppose that \\\B>> and satisfies equation for any \Cyl\B>>, then it satisfies the same equation for every \\>. Suppose that \\>> and satisfies equation for any \Cyl>, then it satisfies the same equation for every \\>. <\proof> See Appendix. <\proposition> Let Cyl\B>>, then there exists a classical solution >>\C\B|)>> to the resolvent equation with the following properties: <\enumerate-roman> >>\\> and moreover, if >\P> for some >\Cyl>, then there exists >>>\\> such that we have >>=>>>\P>. Suppose that has compact support in Fourier variables, then there exists \0> such that, for every ,\\\\B>> and for every \>, there are two constants ,\,\>\0> and 0> such that <\equation*> \G>>\\-\\|)>|\|>\\+|\+K> C,\,\>, where the constant included in the symbol > does not depend on >, >, >, > or>. If \4\|~>>, then there exists 1> such that, for every measurable :B\B\B/2>> and every \\\B>>, we have <\equation*> |\>G>>,\|\>|\|>\\\\>|\|\<\|\|\>>/2>>\\\|)>, uniformly in \0>. Let |~>> be the parameter defined in Remark. In the light of Proposition, the proof of which we postpone to Section, we can then proceed with the proof of uniqueness of solutions to Problem. <\theorem> Let \2\|~>> and take =C>>|)>> and =B>>, where the parameters are taken as in Definition and the space >> is defined as in equation. Then the solution to FPK equation in the sense of Problem is unique. <\proof> Let > and > be two solutions of Problem. We want to show that, for any Cyl\B>> with compact support in Fourier variables, F\ \=F\ \>. This implies =\> since Cyl\B>> with compact support in Fourier variables separates points of the space of Radon measures on \B>. Consider the solution >>> of the resolvent equation given by Proposition and recall that, by pointin Proposition. Since > and > are solutions to the FPK equations we have in particular that equation holds. Therefore, integrating with respect to > (respectively, with respect to>) the resolvent equation equation and subtracting the integral \G>>\\\=0> (respectively,\G>>\\\=0>) which holds by Lemma, we get the relation <\equation*> F\\=\G>> \\-\>G>>-\>|)> \\,j=1,2, where we omitted the dependences on ,Y|)>> for the sake of brevity. Taking the difference of such a relation for yields <\equation*> F\\-F\\=\G>>\\\-\G>>\\\-\>G>>-\>|)> \\+\>G>>-\>|)> \\. Now, pointin Proposition gives us, for any \0>, <\eqnarray*> F\\-F\\|\|>>|>|+>|)>|\+K>+\>G>>-\>|)> \\|\|>+\>G>>-\>|)> \\|\|>>>>> On the other hand, point in Proposition implies that <\equation*> \>G>>-\>|)> \\|\|>\>|\-\>|\<\|\|\>>/2>>\\\|)>. By the proof of Theorem, Proposition, and using the fact that >\=\>, we get that <\equation*> |\-\>|\<\|\|\>>/2>>\\\|)>\0\\0. Thus, if we take the limit \0> and then \0>, we get \ F\\-F\ \|\|>\\>, and since \0> is arbitrary, we get =\>. In the case of Problem a better result, in the sense that \|~>> (cf.Remark), can be achieved exploiting the properties of the resolvent operator proved in Section. <\theorem> Let \2\\>, =C>>|)>> and =B>>|)>>, where the parameters are taken as in Definition. Then the solution to FPK equation in the sense of Problem is unique. <\proof> The proof is the same as in the case of Problem, but substituting point of Proposition, with Proposition below. To solve the resolvent equation as needed in Proposition, \ we use a probabilistic representation and look for solutions in the form <\equation> G>>,Y|)>=\>>et>F>,Y>|)>\\t|]>, where >> and >> are solutions to the stochastic differential system <\eqnarray*> X>>||+m|)>X>+\,>>|Y>>||+m|)>Y>-\>>,Y>|)>,>>>> where >> is defined in equation, > is a Gaussian space-time white noise, and with initial conditions ,Y|)>\B\B>. This is a triangular system, where the first equation does not depend on the second variable. The first equation is a linear SDE with additive noise with stationary solution given by the Ornstein-Uhlenbeck process, whose invariant measure is the Gaussian free field with mass0>. To prove that >>> is a classical solution we need some properties on the flow induced by the SPDEs\U. We will only state such properties in the present section \U postponing their proof to Appendix \U and then proceed by showing that Proposition holds. First, let us write down the equations for the derivatives of the flow. Let us denote by and the solutions to equations\U, dropping the dependence on the parameter \0> for simplicity of notation. The derivatives of solve <\eqnarray*> -\+m|)>\>X>||\>X=id,>>|-\+m|)>\>X>||\>X=0,>>|-\+m|)>\>X>||\>X=0.>>>> Whenever \B>>, <\equation> -\+m|)>\>Y|)>=-\\>,Y|)>>Y|)>|]>,\>Y=h, for B\B>>, and <\eqnarray*> -\+m|)>\>Y|)>>||\>,Y|)>>Y|)>|]>-\\>,Y|)>>X|]>,>>|>Y>||>>> for B>, and <\eqnarray*> -\+m|)>\>Y|)>|]>>||\>,Y|)>>Y|)>|]>|]>>>|||\>,Y|)>>Y|)>,\>Y|)>|]>|]>>>|||\>,Y|)>>X,\>Y|)>|]>|]>>>|||\>,Y|)>>X|)>,\>X|)>|]>|]>,>>|>Y>||>>> for \L|)>>. The derivatives \,|\>>> have the following expressions <\equation*> \\>,Y|)>|]>=\\>,Y|)>|]>=\ f>e\g>|)>>:e\g>|)>>|)>>\\|)>, and <\eqnarray*> \>,Y|)>,\|]>=\\>,Y|)>,\|]>>||\>,Y|)>,\|]>>>||| f>e\g>|)>>:e\g>|)>>|)>>\\|)>>\\|)>.>>>> Hereafter, we will use the notation =\/|)>>, and <\equation> =B\B\|)>-\>r,p\r,\>|)>,=B,\,\>>|)>, where all the appearing parameters are defined as in Definition.\ <\proposition> For any \0>, if ,Y|)>\\\B>|}>>, then there exists a unique solution > to equations\U such that\ <\equation*> ,Y|)>\\\B>|}>,t\\. Let and be solutions to equations\U.\ <\enumerate-roman> For every \0>, we have that that the derivatives >X>, >X>, >X>, >Y>, >Y>, and >Y> of and exist and satisfy equations,,,,, and, respectively. Furthermore, they are all continuous functions with respect to > and >. For every \>, \|)>>, ,\\1>, we have the estimates <\eqnarray*> |||\>Y|)>|\<\|\|\>>>>,C>-2\>|)>|)>\L>,|)>>>>||>>>|||1>|f> \e>\X|)>>:eP>\h|)>>||1>>>>,L>>|)>|)>>,|h|\<\|\|\>>>|>,h\,>>>> and <\eqnarray*> |||\>Y|)>|\<\|\|\>>>>,C>-2\>|)>|)>>>>||>>>||~>||f> \e>\X|)>>:e P>\h|)>>|\<\|\|\>>>>,L>>|)>|)>>|>,h\,>>>> where > and |~>> are two second degree polynomials. For every ,\\0>, we have that there exist ,\\0> such that <\equation> |\>Y|)>|\<\|\|\>>>>,H>>|)>>\>\f>|)> \e>\X|)>|)>>:e>\Y|)>|)>> e-m|)>t>|\|>>|)> \z|)>. The remainder of the present subsection is devoted to the proof of Proposition. <\render-proof|Proof of Proposition> Let us start by proving that >>> is a classical solution of the resolvent equation. We exploit Itô's formula appearing in Theorem4.32 in. Notice that it can be applied to >>> since, by point in Proposition, >>> is a >-function with trace-class second derivative (continuity of the second derivative can be deduced with similar techniques as the ones adopted in Proposition and point in Proposition). We use the notation >> to denote a process at time , with starting point > at time . Recalling the definition of >>> in equation, we have <\equation*> G>>>,Y>|)>=>es> \>>,Y>|)>\|\|]> \s, where |)>0>> is the filtration generated by the white noise at time and the initial conditions > and>. By Markovianity of the process, we get\ <\equation*> G>>>,Y>|)>=>es> \>>,Y>|)> \|\|]> \s. Since from now on and always start at > and >, respectively, when , we drop the dependence on the initial conditions. We have, by Itô's formula, <\eqnarray*> ||t>G>>,Y|)>-G>>,Y|)>>>|||es>\>>>,Y|)>|)> \s+es>\>G>>,Y|)>\\X-\es>G>>,Y|)> \s.>>>> On the other side of the equation, we have <\equation*> >e>\>,Y|)>\|\|]> \s->es>\>,Y|)>\|\|]> \s. Notice that <\equation*> \>e>\,Y|)>\|\|]> \s=\>es>\,Y|)>\|\|]> \s, and therefore <\equation*> \>es>\,Y|)>\|\|]> \s-\>es>\,Y|)>\|\|]> \s=-\es>\,Y|)>\|\|]> \s. Dividing by and letting 0>, we get\ <\equation*> -\es>\,Y|)>\|\|]> \s\-F,Y|)>. Taking expectation also on the first side, we have the desired relation. > When Cyl\B>>, the derivative of >>,Y|)>> with respect to> is givenby <\equation*> \>G>>,Y|)>=\>|>et>|\ |\>Y,v|\>|> \t|>. Therefore, we have <\equation*> >G>>,Y|)>|\|>\sup,N>\>>et> |\|\<\|\|\>>>> |\>Y,v|\>|\|> \t|]> . Applying the estimate, we get <\eqnarray*> |\>Y,v|\>|\|>>|>|> |\>Y|)>|\<\|\|\>>>>,C>-2\>|)>|)>\L>,|)>>|v|\<\|\|\>>>>>>||>|> P||f>\e>\X|)>>:e P>\h|)>>|\<\|\|\>>>>,L>>|)>|)>>,|h|\<\|\|\>>>|>|v|\<\|\|\>>>>.>>>> Therefore, <\eqnarray*> >G>>,Y|)>|\|>>|>||\|\<\|\|\>>>>|v|\<\|\|\>>>>|)> >et> \ \> >>|||\>|P||f>\e>\X|)>>:e P>\h|)>>|\<\|\|\>>>>,L>>|)>|)>>,|h|\<\|\|\>>>|>|> \t.>>>> We have that <\eqnarray*> ||>|P||f>\e>\X|)>>:e P>\h|)>>|\<\|\|\>>>>,L>>|)>|)>>,|h|\<\|\|\>>>|>|>>>||>|>||f>e>\X|\|>>|\<\|\|\>>>>,e sup0> \>\X|)>|]>>,e|h|\<\|\|\>>>>|>,>>>> where >> is a suitable second degree polynomial. Taking the supremum over > with \1>, recalling that >\B\|)>+\>>|)>>, for some |)>>, and taking >>>|)>> as in Definition to be proportional to |f>exp>\X|\|>|)>|\<\|\|\>>>>+1>, we get that >>> satisfies the first condition of Definition. For the remaining conditions of Definition, similar arguments with the application of inequalities and allow us to conclude that >>\\>.\ Since in our approximation we have that >=\ f> e>\|)>-|2>c>>>, then the sum >=X>+Y>> solves the SPDE <\equation*> -\+m|)>\>=-\ f> eg>\\>-|2>c>>+\. Therefore, if Cyl\B>> is of the form >\P> for some >\Cyl>, we have <\equation*> G>>,Y|)>=>>>+Y|)>=\>>et>>>|)>\\t \=X+Y|]>. > In the following, for , we denote by ,j>,Y,j>|)>> the solution to the system of equations\U with initial conditions ,Y|)>\\>. Sometimes we also write ,Y|)>> to indicate that ,Y|)>\\>, for . By definition of >>> (i.e.by equation), we have <\eqnarray*> ||G>>,Y|)>\\X,\Y|)>-\G>>,Y|)>\\X,\Y|)>>>|||>et> \>,1>,Y,1>|)>|]>\\t\\\-\>et> \>,2>,Y,2>|)>|]>\\t\\\>>|||>et>\,1>,Y,1>|)>-F,2>,Y,2>|)>|]> \t,>>>> Notice that, for any \0>, there exist two compact sets ,K>, such that |)>,\|)>\1-\>. Therefore, the following inequality holds <\eqnarray*> ||>et>\,1>,Y,1>|)>-F,2>,Y,2>|)>|]> \t|\|>>>||>|>et>\,1>,Y,1>|)>-F,2>,Y,2>|)>|\|> \ \,Y|)>\K,,Y|)>\K>|]> \t|\|>+2\|F|\<\|\|\>>>>\>e\>\\.>>>> Let us then focus on the case where ,Y|)>\K> and ,Y|)>\K>. By Lagrange's theorem, we have\ <\equation*> F,1>,Y,1>|)>-F,2>,Y,2>|)>=|\F>,\>|\>\\\, where, for \>, <\equation*> \F>=\F>|)>X+\X|)>,Y>|)>Y+\Y|)>|)>, with F=\|u,X|\>+\|v,Y|\>>, for Cyl\B>> having the following form =|u,X|\>,\,|u,X|\>,|v,Y|\>,\,|v,Y|\>|)>>, with \C|)>> and \\|)>>, ,N>, and \\|)>>, ,M>, and, for \>, <\equation*> \>=-X>X>|)>X+\X|)>,0,\-X>Y>|)>Y+\Y|)>,\-Y>Y>|)>Y+\Y|)>|)>. We want to get some bounds for |\F>,\>|\>\\|\|>>.\ For the term |\F>,\-X>X>|)>X+\X|)>|\>|\|>> we have\ <\eqnarray*> |\F>,\-X>X>|)>X+\X|)>|\>|\|>>|||\F>,P|)>X+\X|)>|\>|\|>>>||>|t>|\F>|\<\|\|\>> ||)>X+\X|\<\|\|\>>>>>.>>>> For the term |\F>,\-Y>Y>|)>Y+\Y|)>|\>|\|>>, considering equation, and multiplying it by >\\>Y> and exploiting the negativity of the non-linearity, we get the following a priori estimate for some constant |)>> <\equation*> \|>\\-Y>Y>|)>Y+\Y|)>|\<\|\|\>>>+k|>\\-Y>Y>|)>Y+\Y|)>|\<\|\|\>>>\0, and therefore, by Gronwall lemma, <\equation*> |>\\-Y>Y>|)>Y+\Y|)>|\<\|\|\>>>\e, and the estimate is independent of>. Consider now the operator >\f>, and recall that in Fourier representation convolution corresponds to multiplication, i.e.the previous operator can be viewed as \|^>>\>. When > has compact support, such an operator is invertible if \\\> for some positive constant > depending on the size of the support of >, with inverse given by <\equation*> |^>>:\|^>>>\, which is a well-defined operation because |^>>=|^>k|)>>, |^>=1>, and |^>> is smooth since > is a Schwartz function. Using the previous notation, we have <\eqnarray*> |\F>,\-Y>Y>|)>Y+\Y|)>|\>|\|>>|||\F>\\>,>\\-Y>Y>|)>Y+\Y|)>|\>|\|>>>||>||\F>\\>|\<\|\|\>>>>||>|sup\\\>|\F>\\>|\<\|\|\>>,>>>> where we used the fact that has compact support in Fourier variables and so the norm |\F>\\>|\<\|\|\>>> is bounded for any \\\>, and also that <\equation*> sup\\\>>|supp>>>>>> |^>>|\|>\+\. \; Now, focus on the term |\F>,\-X>Y>|)>Y+\Y|)>|\>|\|>>. By equation, we have <\eqnarray*> |||\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>+|\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>>>||||\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>>>||>|>\\>>|)>X+\X|)>,Y>|)>Y+\Y|)>|)>\-X>X>|)>X+\X|)>>>|||\-X>Y>|)>Y+\Y|)> \z,>>>> and by Hölder and Young inequalities, and re-absorbing the terms properly, we have that, for any \0>, the following inequality holds <\eqnarray*> |||\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>+k|\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>>>||>|> |\\>>|)>X+\X|)>,Y>|)>Y+\Y|)>|)>|\<\|\|\>>> |\-X>X>|)>X+\X|)>|\<\|\|\>>>.>>>> Now, we can bound the norm |\-X>X>|)>X+\X|)>|\<\|\|\>>>> by a constant since > and > belong to compact sets. Moreover, by Lemma, we have <\eqnarray*> |||\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>+k|\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>>>||>|t>|t+s>> |\\>>|)>X+\X|)>,0|)>|\<\|\|\>>>.>>>> Applying Gronwall lemma yields the following inequality <\equation*> |\-X>Y>|)>Y+\Y|)>|\<\|\|\>>>\e|)>\>|\+s>> |\\>>>|)>X+\X|)>,0|)>|\<\|\|\>>>\\. Thus, we get, for some constant >\0> depending on \0>, <\eqnarray*> |||\-X>Y>|)>Y+\Y|)>|\<\|\|\>>> \,Y|)>\K,,Y|)>\K>|]>>>||>||\-X>Y>|)>Y+\Y|)>|\<\|\|\>>> \,Y|)>\K,,Y|)>\K>|]>>>||>|>|e|)>\>|\+s>>\|\\>>>|)>X+\X|)>,0|)>|\<\|\|\>>>|]>\\|>>>||>|>|e|)>\>|\+s>>\\|>,>>>> since the law of >>|)>X+\X|)>> is that of a Gaussian free field independent of \> and of\0>. Putting everything together, we get <\eqnarray*> ||,1>,Y,1>|)>-F,2>,Y,2>|)>|)> \,Y|)>\K,,Y|)>\K>|]>>>||>|t>|\F>|\<\|\|\>>\||)>X+\X|\<\|\|\>>>>>|]>+e|\F>|\<\|\|\>>+C>|e|)>\>|\+s>>\\|>>>||>|t>|\F>|\<\|\|\>>+e|\F>|\<\|\|\>>+C>|e|)>\>|\+s>>\\|>,>>>> where we used that the law of |)>X+\X> does not depend on\>.\ Therefore, we get <\eqnarray*> ||G>>,Y|)>\\X,\Y|)>-\G>>,Y|)>\\X,\Y|)>|\|>>>||>|>t>et>|\F>|\<\|\|\>>+esup\\\>|\F>\\>|\<\|\|\>>>|)>|1+|)>\>|\+s>>\\|>|]>\t|\|>>>||||F|\<\|\|\>>>>\>e\>\\.>>>> For some constant 0>, we have <\eqnarray*> ||>t>et>|\F>|\<\|\|\>>+esup\\\>|\F>\\>|\<\|\|\>>>|)>|1+|)>\>|\+s>>\\|>|]>\t|\|>>>||>|>|)>\>et-K t> \t>>||>|>|)>|\+K>,>>>> and the last term converges to zero as \0>. Thus, we have <\equation*> G>>,Y|)>\\X,\Y|)>-\G>>,Y|)>\\X,\Y|)>|\|>\\+>|)>|\+K>. > Let \B/2>> and note that >Y>|]>> solves equation. We write <\equation> \\>=\>Y>|]>-P\. Recall that in the proof of point in Proposition we saw that\ <\equation*> \>G>>,Y|)>=\>>et>\F,Y>|)> \>Y> \t|]>. Now, recalling the representation of \>>, we have <\eqnarray*> ||\>>et>\F,Y>|)> \>Y>|]> \t|]> \\>>||>||\F|\<\|\|\>>\>>et>|\>Y>|]>|\<\|\|\>>/2>> \t|]> \\>>||>|\>>et>|P\|\<\|\|\>>/2>>+|\\>|\<\|\|\>>>|)>>|)> \t|]> \\.>>>> By Lemma below, we have <\eqnarray*> ||\>>et>\F,Y>|)> \>Y>|]> \t|]> \\>>||>|\>>et>>|f> \e>\X|)>>:|\<\|\|\>>>,B/2>|)>>|)>|\|\<\|\|\>>/2>> \t|]> \\.>>>> Hölder inequality with respect to all measures, together with the fact that \0>, yields <\eqnarray*> ||\>>et>\F,Y>|)> \>Y>|]> \t|]> \\>>||>>||\|\<\|\|\>>/2>> \ \\|)> \>|f> \e>\X|)>>:|\<\|\|\>>>,B/2>|)>>|]> \\|)>>et>>|)> \t.>>>> Here, since \0>, the integral >et>>|)> \t> is finite. Moreover, by Proposition, we have that the term <\equation*> \>|f> \e>\X|)>>:|\<\|\|\>>>,B/2>|)>>|]> \\|)> is bounded with respect to \0>. This concludes the proof of Proposition. We close the section with an auxiliary lemma used in the previous proof. <\lemma> Let \4\|~>>. Then, for every \0>, \B>>, and > in a set of full measure with respect to the free field measure > with mass 0>, we have that <\equation*> |\\>|\<\|\|\>>>|)>>\>|f> \e>\X|)>>:|\<\|\|\>>>,B/2>|)>>|\|\<\|\|\>>/2>>. <\proof> Let =\/|)>\|~>>. The proof is similar to the one of Theorem. Indeed, by multiplying the equation for \>> by >\\\>>, and integrating, we get <\eqnarray*> ||>|>\\\>|\<\|\|\>>>>+\>|>\\\>|\<\|\|\>>>>\\s>>|||\>\>\f>:e>\X|)>>:e>\Y|)>>>\\\>|)>\\s\\z>>||>|\>>f>:e>\X|)>>:e>\Y|)>>>\e+m|)>s>\|)>|)>g>\\\>\\s\\z.>>>> In the last line, we have <\equation*> \> \> f> >\X|)>>:|\>,B/2>|)>|)>> >\Y|)>>|\>>\\|)>> >\e+m|)>s>\|)>|\>>>\,B-\+2\-2/q>/2>|)>|)>> >\\\>|\>/2>>,L>|)>|)>\>>|L/2>,H>|)>|)>>>>>>>\\s. Now, whenever \|~>>, there exist 1>, \\1>, \\1>, \0>, such that the following system has a solution <\equation*> +2\-\0,>>|-\+\\0,>>|+\1,>>|+\-\+|2>\1.>>>>> The proof is complete. > In the case of Problem a better result can be obtained. It is useful to introduce a new approximating operator ,|\>>> for > of the form <\equation*> \,|\>>|)>\ tr\|)>-|+m|)>X,\\|\>-|+m|)>Y+\,|\>>,\\|\>, where <\equation*> \,|\>>\\ f>>\X|)>>:e|\>>\Y|)>>|)>. Here >> and >> are defined as in Section. Recall that by Remark, since |\>,\\0>\,|\>>=\>, Problem with the operator ,|\>>> is equivalent to the one with operator >> defined in equation. We can then consider the resolvent equation associated to ,|\>>>, namely <\equation*> -\,|\>>|)>G>,|\>>=F, where Cyl\B>> with compact support in Fourier variables. A solution to such equation is given by\ <\equation> G>,|\>>,Y|)>=\>>et>F,Y,|\>>|)> \t|]>, where ,Y,|\>>> solves the following system of equations <\eqnarray*> -\+m|)>X>||,X=X,>>|-\+m|)>Y,|\>>>||,|\>>,Y,|\>>|)>,Y,|\>>=Y.>>>> It is easy to show that all the results in Proposition hold also for equations\U, adapting the form of the equations for the derivatives. This implies that points , stated in Proposition hold true also for>,|\>>>. As far as pointin Proposition is concerned, we can get a slightly better result in the present scenario. Recall that > is defined as in Remark. Notice that the operator and equations\U can be defined also for the case |\>=0>. Moreover, (a suitable adaptation of) Proposition holds also for the case |\>=0>, and point of Proposition holds also for>,0>>. <\remark> It is worth to note that the operator ,|\>>> cannot be use directly to solve Problem, since the solution to the resolvent equation depends in general on > and not only on . <\proposition> Let Cyl\B>> and consider ,|\>>>> given by. If \4\\>, then there exists 1> such that, for every \\\B>>, we have <\equation*> lim|\>\0>|\>G>,|\>>,\-\,|\>>|\>|\|>\\\\>|\-\,0>|\<\|\|\>>/2>>\\\|)>, uniformly in \0>. In order to prove the previous result, we need some technical lemmas. First, we deal with the convergence as|\>\0>. In particular, whenever an object has |\>> as one of its parameters (e.g.>G>,|\>>>), the same notation with |\>=0> indicates that it is the limiting object as |\>\0> (e.g.>G>,0>=lim|\>\0>\>G>,|\>>>), whenever it exists. <\lemma> For every \0>, \B>>, and > in a set of full measure with respect to the free field measure > with mass 0>, we have that >Y,|\>>-\,|\>>|)>> converges to >Y,0>-\,0>|)>> in>, as |\>\0>. <\proof> Recall that >Y,|\>>-\,|\>>|)>> solves equation. This means that <\equation*> \\,|\>>=\>Y,|\>>-\,|\>>|)>-P-\,|\>>|)> solves the equation <\eqnarray*> |-\+m|)>\\,|\>>>||\,|\>>,Y,|\>>|)>\,|\>>|]>-\\,|\>>,Y,|\>>|)>-\,|\>>|)>|]>,>>|\,|\>>>||>>> By Corollary, we get the estimate <\equation*> |\\,|\>>|\<\|\|\>>\\>P||f>\e >\X|)>>:|\<\|\|\>>>>,|\-\,|\>>|\<\|\|\>>>|>. Such a bound is uniform with respect to|\>\0> and therefore we get a converging subsequence whose limit\,0>> solves equation with |\>=0>. Since the term -\,|\>>|)>> converges point-wise to -\,0>|)>>, the result is proved. Let us prove that it makes sense to consider some parameters satisfying certain conditions that will be useful in upcoming results.\ <\lemma> Suppose that =\/|)>\\>. Then there exist 1> such that q\2>, \0>, and and > as in Definition, such that the following inequalities are satisfied <\equation> -\-\-2\+\\0,+\1,|2>q\1. <\proof> In order to prove the result, it is enough to show that there exists some solution with the previous properties to the system of equations <\equation*> -\-\+|r>=0,+=1,|2>q=1. From such relations we get the equality <\equation> \=|r+1|)>>. We get the maximum value > of > for some >\2.52>. Since, with this choices of parameters, we have 1.21>, the result is proved. In the proof of Lemma, we introduced the object <\equation> \\,|\>>=\>Y,|\>>-\,|\>>|)>-P-\,|\>>|)>, satisfying equation and admitting a limit \,0>> as |\>\0>. <\lemma> Suppose that \4\\>, consider the parameters and defined as in Lemma, and ,\> such that q/2\2>, r/2\2> and r\1>. Then, for every \0>, \B>>, and > in a set of full measure with respect to the free field measure > with mass 0>, we have that <\equation*> |\\,0>|\<\|\|\>>>|)>>\>|f> \e>\X|)>>:|\<\|\|\>>>,B/2>|)>|)>>|\-\,0>|\<\|\|\>>/2>|)>>. <\proof> Let =\/|)>>. With the same notation as in Lemma, we consider the limit \,0>> and show that it converges to zero. Indeed, it satisfies <\equation*> -\+m+\f> \e>\X|)>>:eY>|)>\\,0>=-\f> \e>\X|)>>:eY>P-\,0>|)>, with \,0>=0>.\ We now want to exploit a similar argument as the one used in Theorem to get some estimates concerning the solution to the previous equation. Let \\> be an increasing smooth function with bounded derivatives and such that =0>, \\1> as \\>, and define =w \y>. Then, multiplying the equation by >\|)>w|~>,0>|)>>, where ,\\0>, and integrating, we have <\eqnarray*> |||\>\|)>W\,0>|)>|\<\|\|\>>>+m>|\>\|)> w\,0>|)>|~>,0>|\<\|\|\>>>+|\>\|)> w\,0>|)>\,0>|)>|)>|\<\|\|\>>>>>|||\\>z|)> f> \e>\X|)>>:eY> w\,0>|)>\\,0> \z>>||>|\\/2>z|)>f> \e>\X|)>>:eY> w\,0>|)> \/2>z|)>-\,0>|)>|)> \z,>>>> where m>\-\>z|)>|\>z|)>>|\|>>> for every \>, which holds for \0> small enough.\ If we choose the parameters as in Lemma, we have <\eqnarray*> /2>\|)>f> \e>\X|)>>:eY>>|>|>,B|)>,>>|\,0>|)>>|>|>|)>,>>|/2>\|)>-\,0>|)>|)>>|>|>|\|>/2>\,B-\+\>|)>|)>.>>>> Therefore, by integrating with respect to time equation and recalling that \,0>=0>, we get <\eqnarray*> |||\>\|)>W\,0>|)>|\<\|\|\>>>\>>||||\/2>\|)>f> \e>\X|)>>:eY> w\,0>|)>|\<\|\|\>>> |\/2>\|)>+m|)>s>-\,0>|)>|)>|\<\|\|\>>-\+\>> \s>>>> where we used the fact that +\1> and -\-2\+\\1>. By Proposition, we have that <\equation*> |f> \e>\X|)>>:eY> w\,0>|)>|\<\|\|\>>/2>>\|f> \e>\X|)>>:|\<\|\|\>>/2>>|eY>|\<\|\|\>>>>|w\,0>|)>|\<\|\|\>>>>. Therefore, by Proposition, we have <\equation*> |\>\|)>W\,0>|)>|\<\|\|\>>>\|f> \e>\X|)>>:|\<\|\|\>>/2>> s/2>|\-\,0>|\<\|\|\>>/2>> \s. Multiply and divide then by >> and apply Hölder inequality with respect to to get <\eqnarray*> |||\>\|)>W\,0>|)>|\<\|\|\>>>>>||>||\/2>\|)>f> \e>\X|)>>:|\<\|\|\>>>,B/2>|)>>|\-\,0>|\<\|\|\>>/2>>s/2>\q> \s|)>.>>>> Since the expression s/2>\q> \s|)>> is bounded for 0> and grows polynomially in time as +\>, we deduce that the result holds. <\render-proof|Proof of Proposition> As in the proof of point of Proposition we can get\ <\equation*> \>G>,|\>>,Y|)>=\>>et>\F,Y,|\>>|)> \>Y,|\>> \t|]>. Therefore, by Lemma and Lebesgue's dominated convergence theorem, we have <\eqnarray*> |\>\0>\>G>,|\>>,Y|)>-\,|\>>|)> \\>|||\>\0>\>>et>\F,Y,|\>>|)> \>Y,|\>>-\,|\>>|)> \t|]> \\>>|||\>>et>\F,Y,0>|)> \>Y,0>-\,0>|)> \t|]> \\>>|||\>G>,0>,Y|)>-\,0>|)> \\.>>>> Now, recalling the representation of \,0>>, we have <\eqnarray*> ||\>>et>\F,Y,0>|)> \>Y,0>-\,0>|)> \t|]> \\>>||>||\F|\<\|\|\>>>>\>>et>|\>Y,0>-\,0>|)>|\<\|\|\>>/2>> \t|]> \\>>||>|\>>et>|P-\,0>|)>|\<\|\|\>>/2>>+|\\,0>|\<\|\|\>>>|)>>|)> \t|]> \\.>>>> At this point, the proof follows the same steps as the one of pointin Proposition. In this section, we prove existence of solutions to Problem and. Since existence for the Problem implies existence for Problem (see point in Theorem), we only prove the first. We actually prove a stronger statement than the one given in Problem, namely that equation holds for every \\> and not only for \Cyl\B>>. We exploit a strategy based on Lyapunov functions.\ Let us introduce a finite-dimensional approximation to the operator>. Let \>, \0>, and consider > the two-dimensional torus of size which we identify hereafter with the subset ,M\|]>\\>. Consider the :\|)>\C>|)>> definedas, for \|)>>, <\equation*> Q=Fej\F,Fej=|\|>,|\|>\N-1>|\|>|N>|)>|\|>|N>|)> e,x\\. Let us stress that the operator > is both a positive operator, i.e.|Q,F|\>\0>, and a positive preserving operator, namely, if is a positive distribution, then > is a positive function. The latter property following from the positivity of the kernel>. The new approximating operator >> is then givenby, for \\>, by <\eqnarray*> >|)>>|| tr|)>> >>\\|)>P>|)>-|+m|)>X,\\|\>>>||||+m|)>Y+\>,\\|\>,>>>> where >> is the following approximation for the non-linearity in equation, <\equation> \>=\ Q>\eQ>\X|)>>:eQ>\Y|)>>|)>|)>. Here, >> is defined as in Section, >> is the natural projection of |)>> on the space|)>>, and >> is the periodicization on the torus>. Moreover, the term expQ>\X|)>|)>:> is defined as follows <\equation> \eQ>\X|)>>:=eQ>\X|)>>-|2>c>, where >=>Q>|)> +m|)>g>|)> \z>, where the inverse +m|)>> is taken with periodic boundary conditions on>. The system of equations for the flow is then givenby <\eqnarray*> -\+m|)>X>||,>>|-\+m|)>Y>>||>,Y>|)>,>>>> where >> is negative, periodic, and it belongs to a subspace of \span/M>;\N|}>>. As usual, we drop the dependence on the parameters > when no ambiguity occurs. We introduce some Lyapunov functions >, >, and > that will be crucial in the proof of the estimates for and. In particular, we take them such that > and > depend on both and , while > depends on only. <\lemma> Consider two Banach spaces > and >. Let |~>> be an operator taking values in some space of functions> and consider a measure |~>> such that |~>\\\|~>=0> for every \\>. Suppose that there exist some functions ,V:B\B\\>, and :B\\>, such that we have ,V,V\\> and <\enumerate-roman> > and > are positive, The inequality <\equation*> |~>V\-V+V holds true. Then, we have <\equation> V |~>X,\Y|)>\V |~>X,\Y|)>. <\proof> The statement follows from the fact that |~>V |~>X,\Y|)>=0>. We now work on the estimate for . We will have to choose the aforementioned Lyapunov functions, we start with :B\\>, that we take to be, for \0>, <\equation*> V=|X|\<\|\|\>>>>>. Note that, since corresponds to the free field, the previous norm is finite. Notice also that the previous function is not cylindric. Moreover, if we consider the representation of Besov spaces given in Proposition, then we have, for /2+1\0>, <\equation*> |X|\<\|\|\>>>>>\|\|)>X|\<\|\|\>>>>+>>>|\|>|t/2+1>> X|\|> \z \t\V+V, where is to be determined, similarly as in the proof of Proposition. <\proposition> Let \0>, 1> and \0> such that \2> and /2+1\0>. Consider to be the solution to equation on the torus>. For some \>, we have the inequality <\equation> \>V=\>|X|\<\|\|\>>>>>|)>\-|)>|X|\<\|\|\>>+2/p>>>+>C. <\proof> We want to evaluate >V>. The gradient of > is givenby <\eqnarray*> V>|||\|)>X|\<\|\|\>>>>+>>>|\|>|t/2+1>> \X|\|>|)> \z \t>>|||>>|\|>|)>X|\|>sign|)>X|)>\|)>h \z>>|||>>>|\|>|t/2+1>> X|\|>signX|)> Ph \ \z \t,>>>> while its second derivative is <\eqnarray*> ||V|)>>>|||>>|\|> \|)>X|\|> sign|)>X|)> \|)>h|)>|)> \z>>|||>>>|\|>|t/2+1>> \X|\|> signX|)> Ph|)>|)> \ \z \t>>|||>>|\|>|)>X|\|> |)>h|)>|)>h|)> \z>>|||>>>|\|>|t/2+1>> X|\|> h|)> h|)> \ \z \t.>>>> The first term to deal with is |+m|)>X,\V|\>>. We have\ <\eqnarray*> |+m|)>X,\V|\>>||>>|\|>|)>X|\|>sign|)>X|)>|)>+m|)>X|)> \z>>|||>\>|\|>|)>X|\|>sign|)>X|)>|)>\|)>X|)> \z>>|||>>|\|>|)>X|\|> \z>>|||>>|\|>|)>X|\|>|)>X|)>|\|> \z>>|||>>|\|>|)>X|\|>\>|)> sign|)>X|)>\|)>X|)> \z>>|||>>|\|>|)>X|\|> \z.>>>> Rearranging the second term on the right-hand side and exploiting Young's inequality, we have, for all \0>, <\eqnarray*> |+m|)>X,\V|\>>|>|>>|\|>|)>X|\|>|)>X|)>|\|> \z>>|||>>|\|>|)>X|\|>\>|\|>|\>> |)>X|\|>/2>|)>X|)>|\|> \z>>|||>>|\|>|)>X|\|> \z>>||>|>>|\|>|)>X|\|>|)>X|)>|\|> \z>>||||4\>>>|\|>|)>X|\|>\>|\|>|\>> \z>>|||\> >|\|> |)>X|\|> |)>X|)>|\|> \z>>|||>>|\|>|)>X|\|> \z,>>>> from which, choosing \0> small enough and 0> large enough, and appropriately rearranging the terms, we get <\equation*> -|+m|)>X,\V|\>\-|\|)>X|\<\|\|\>>>>. We also have <\eqnarray*> |||+m|)>X,\V|\>>>|||>>>|\|>|t/2+1>> X|\|>signX|)>+m|)>X|)> \z \t>>|||>>>|\|>|t/2+1>> \X|\|> \z \t>>|||>>\>|\|>|t/2+1>>|)> X|\|> \z \t>>|||>>|2>-1|)> >|\|>|t/2+2>> X|\|> \z \t>>|||>>|2>-1|)> >|\|>|t+2/p|)>/2+1>> X|\|> \z \t>>||>|||X|\<\|\|\>>+2/p>>>-|\|)>X|\<\|\|\>>>>|>.>>>> The second term to deal with is the trace of the second derivative, i.e.|)>>V|)>>. By the results presented in SectionXI.V of, it suffices to consider the second derivative with =h>, i.e.V>, and integrate with respect to. Then, exploiting Young inequality for products and using the fact that \2>, we have for any \0> <\eqnarray*> V|)>>||>>>|\|>|)>X|\|> |^>|\|> \z \y>>|||>>|\|>|)>X|\|> >|^>|\|> \y \z >>||>|>>|\|>|)>X|\|> \z >>||>|> p>>|\|> \z+\p>>|\|>|)>X|\|> \z >>||>|> C+\|\|)>X|\<\|\|\>>>>.>>>> We get, if > is the heat kernel on the torus>, that, for any \0>, <\eqnarray*> V|)>>||>>>>|\|>|t/2+1>> X|\|> |\|> \ \z \t \y>>|||>>>|\|>|t/2+1>> X|\|>> |\|> \y|)> \ \z \t>>|||>>>|\|>|t/2+1>> X|\|>t>|t> \ \z \t>>|||>>t>>|\|>|t/2+2>> X|\|> \ \z \t>>||>|>C>>>|\|>emt>|t/2+2>> \ \ \z \t+\>>>|\|>|t/2+2>> X|\|> \ \z \t>>||>|>C+\||X|\<\|\|\>>+2/p>>>-|\|)>X|\<\|\|\>>>>|>,>>>> where we used Young inequality as above.\ Choosing \0> small enough, we get <\equation*> \V\-|)>|X|\<\|\|\>>+2/p>>>+>C, which gives the result. <\remark> The result in Lemma with =B>, no dependence on > whatsoever, and with Lyapunov functions chosenas <\eqnarray*> >|||X|\<\|\|\>>>>>,>>|>|||)>|X|\<\|\|\>>+2/p>>>,>>|>|>|>C,>>>> for some constant 0>, gives the estimate <\equation*> |)>|X|\<\|\|\>>+2/p>>> \>X,\Y|)>\>C. Notice further that the bound can be chosen to be uniform with respect to the size of the torus> since the integrals >|^>|\|> \y> and > |\|>|\|> \y> are uniformly bounded in. It is worth to note also that by the Besov embedding (Proposition) the space +2/p>>|)>> is embedded in =C>>|)>>. Let us choose the Lyapunov functions in order to get an estimate for . We may need V|)>> to be > and V,Y|)>> to be>.\ Let 0>. Referring to the representation of Besov spaces given in Proposition and Remark, take p\1>, and consider <\equation> V=|Y|\<\|\|\>>>>\|Y|\<\|\|\>>>>+>>>|\|>|t> >PY|\|> \z \t\V+V. <\theorem> Suppose that is a solution to equation. Let =\/|)>\2>, and p,q\+\>. Take and such that <\equation> 0\s\\+2->,-s--2|)>-8\>|2\>\r\-s+-2|)>-8\>|2\>,\r\2. If X>:\B>=B-\>>|)>>, for > large enough, then we have, for any \0>, <\equation> \>V=\>|Y|\<\|\|\>>>>|)>\-|)>|Y|\<\|\|\>>>>+> |\>|\<\|\|\>>/|)>>>>. <\proof> In the following proof, we neglect the term > appearing in, which can be dealt with in a similar way as in the proof of Proposition. The gradient of > is givenby <\eqnarray*> V>||>>>|\|>|t> \>PY|\|>|)> \z \t>>|||>>>|\|>|t> p >PY|\|> \>Ph \z \t.>>>> Therefore, taking +m|)>Y-\>>, we have <\eqnarray*> |||+m|)>Y+\>,\V|\>>>|||>>>|\|>|t> p>PY|\|> \>P+m|)>Y-\>|)> \z \t>>|||>>>|\|>|t> \>PY|\|>|)> \z \t>>|||>>>|\|>|t> p>PY|\|> \>P>|)> \z \t>>||>|+I.>>>> Let us focus on>. Integrating by parts, we get <\eqnarray*> >||>>\>|\|>|t>|)> >PY|\|> \z \t>>|||>>-1|)> >|\|>|t> >PY|\|> \z \t>>|||>>-1|)> >|\|>|t/2+1>> >PY|\|> \z \t>>||>|||Y|\<\|\|\>>>>-|Y|\<\|\|\>>>>|> .>>>> Consider now>. Exploiting Hölder inequality with ,p> and introducing \0> yields <\eqnarray*> ||>||>>>|\|>|t> p>PY|\|> \>P>|)> \z \t>>|||>>k>>|\|>|t/2>> p>PY|\|>|]> >>|||/2> >|\|> \>P>|)>|]> \z \t>>||>|>> \ k>>|\|>|t/2>> p>PY|\|>|\|>> \z t|t>|)>>>>|||>> \ /2>>|\|>|)> \>P>|)>|\|>> \z t|t>|)>>.>>>> If we apply Young inequality, we then have, for any \0>, <\eqnarray*> |>|>|>> \ k>>|\|>|t/2>> p>PY|\|>|\|>> \z t|t>|)>/p>>>|||> >> \ /2>>|\|>|)> \>P>|)>|\|>> \z t|t>|)>/q>>>|||>> \ k>>|\|>|t/2>>|)>> p>>PY|\|>p-p> \z t|t>|)>/p>>>|||>>> \ /2>>|\|>|)> \>P>|)>|\|>> \z t|t>|)>/q>>>|||>> \ k>|t/2+sp/2>> |t|)>/|)>>>|)>> \ p>|\>>>||||>|\|>>>PY|\|>> \z t|t>|)>/p>>>|||>>> \ /2>>|\|> \>P>|)>|\|>> \z t|t>|)>/q>>>||||Y|\<\|\|\>>|)>/>,p,\>>/p>+>>> \ /2>>|\|>|)> \>P>|)>|\|>> \z t|t>|)>/q>>>||>|+I.>>>> We want to apply Besov embedding (see Proposition) in order to reabsorb the term > in>: considering the parameters of the involved norms, we get the condition <\equation> s\|p-1>->, so that the reabsorbing procedure follows by a suitable choice of>. On the other hand, we have by Proposition <\eqnarray*> >|>|>|\>|\<\|\|\>>,q,\>>>/q>.>>>> All in all, we have the inequality\ <\equation*> \>V=\>|Y|\<\|\|\>>>>\-|)>|Y|\<\|\|\>>>>+>|\>|\<\|\|\>>,q,\>>>/q> , provided condition holds true and > is chosen in an appropriate way.\ Now, in order to obtain, we consider =r> and \\>, and have to check the condition <\equation*> \r+r-2|)>+2\0, which is satisfied with the choice of parameters given in. <\remark> If 0>, we have the bound <\equation*> |\>|\<\|\|\>>/|)>>>>\|\>|\<\|\|\>>/|)>>>>, on the right-hand side of inequality. In that case, choosing the Lyapunov functions as follows <\eqnarray*> >|||Y|\<\|\|\>>>>,>>|>||>|Y|\<\|\|\>>>>,>>|>||>|\>|\<\|\|\>>/|)>>>>,>>>> \ we get that the conditions in Lemma are satisfied thanks to Theorem. Therefore, we have <\equation*> |Y|\<\|\|\>>>> \>X,\Y|)>\>|\>|\<\|\|\>>/|)>>>> \>X,\Y|)>. Equations and induce a Markov process on \B>>. This is due to the fact that, if we start from a negative initial condition for , then remains negative throughout all the evolution. <\proposition> \ The operator >,\|)>> is the restriction of the generator of the Markov process associated to equations and to the space of functions>. <\proof> The proof is based on the fact that we can apply Itô formula to the functions in>. Exploiting Itô formula, the proof follows the argument of the proof of point in Proposition. By linearity, the invariant measure for equation is given by the free field measure with mass on the torus>, that we can identify as usual with the periodic free field of mass on the whole space>. We are interested in studying (infinitesimally) invariant measures for the aforementioned pair of equations, i.e.equations and. Indeed, by Proposition, an invariant measure for those two equations is a solution to a FPK equation associated with >> in the sense of Problem. Let us start off by giving an argument for the existence of an infinitesimally invariant measure (for its definition and the relation with invariant measures see Chapter5 in) by means of the Lyapunov functions introduced in Section. <\proposition> Equations and admit an infinitesimally invariant measure. <\proof> The proof follows the same argument as Lemma3.3 in. We start equations and from deterministic initial conditions ,y|)>\B\B>>, and let ,y|)>,t>> denote the measure of the solution of the equations starting at ,y|)>>. Let <\equation*> |~>=T\,y|)>,t> \t. If such a measure is tight with respect to, and |~>> is its weak limit as +\>, then |~>> is an infinitesimal invariant measure of the equation. We consider the following Lyapunov functions, consisting of the sum of the one considered in Section where we dealt with and separately (see Remarks and), namely <\eqnarray*> >|||X|\<\|\|\>>>>>+|Y|\<\|\|\>>>>,>>|>|||)>|X|\<\|\|\>>+2/p>>>+C>|Y|\<\|\|\>>>>,>>|>||>|C+|\>|\<\|\|\>>/|)>>>>|>.>>>> Then, by Proposition and Theorem, we have <\eqnarray*> ,Y|)>-V,y|)>|]>>||\>V>,Y>|)> \\>>||>|\>,Y>|)>+V>|)>|]> \\.>>>> Recall that >=e+m|)>\>x+>e+m|)>-s|)>>\ \s>, then ||\>>,0|)>|\<\|\|\>>/|)>>>>|>> converges exponentially to a constant, and hence <\equation*> K=sup|)>>\|\>>,0|)>|\<\|\|\>>/|)>>>>|]>\\\+\. Therefore, we have, for any 0>, <\equation*> \,Y|)>-V,y|)>|]>+\>,Y>|)>|]> \\\K, which yields <\equation*> \>,Y>|)>|]> \\\K+\,y|)>|]>. Taking the supremum over |)>>, we have <\equation*> sup|)>> \>,Y>|)>|]> \\\+\. On the other hand, for any 0>, <\equation*> V |~>X,\Y|)>=\>,Y>|)>|]> \s. Since > has compact sub-levels on \B>>, tightness of|~>> is implied. We can actually prove a stronger result on the form of one invariant measure for equations and. Indeed, if we start from =0> and =0> in the proof of Proposition, then we have that the invariant measure that has been built there must be the measure of the process solving the equation <\equation> X=>P\ \s,Y=>P\>,Y|)> \s. It is possible, in fact, to prove that the previous integral equations admit a unique solution.\ <\proposition> Equation admits a unique solution in \\\B\B>. Moreover, for every \>, the law of ,Y|)>> is a solution to the FPK equation for >> in the sense of Problem. <\proof> Existence and uniqueness for equation can be proved in the same way as in the proof of Theorem. In order to prove the second part of the result, we exploit the fact that\ <\equation*> law,Y|)>=lim+\>\,t|)>>=lim+\>\,s|)>> \s. Indeed, if > and > are the processes solution to equations and starting at time with initial condition zero, then we have that <\equation*> law,Y|)>=\,t|)>>, and that ,Y|)>\,Y|)>> as +\>, almost surely. Therefore, by the proof of Proposition, the law of ,Y|)>> is an invariant measure for equations and, and thus a solution to the FPK equation for >>. Hereafter, we denote by >> the law at fixed time of the process ,Y|)>> solution to equation, which is a solution to the FPK equation for >>. Thanks to the representation given by Proposition, we are able to establish some more precise properties of the measure>>. <\proposition> For every \>, and every \0>, the measure >> satisfies the following properties <\enumerate-roman> >|)>\B>|)>\\>, For every \>, we have <\equation*> \>F \\>=0, For every \>, we have <\equation*> \>F G \>X,\Y|)>=F\>G \>X,\Y|)>. <\proof> Property follows from the fact that the solution > to the second equation in is supported on the image of the projection of the operator >, which is exactly>.\ Property is due to the fact that >> solves the FPK equation for>>. Indeed, the system can be split in an infinite-dimensional linear equation and an independent finite-dimensional non-linear equation. The statement then follows from the results in Section2.3 in for the infinite-dimensional part and Theorem 5.2.2 in for the non-linear finite-dimensional part. As far as point >is concerned, letting =X+Y> and calling > the projection of > on > and > the projection on >>, we have that > and > solve two independent equations: the equation for > is a non-linear finite-dimensional equation, since > is a finite-dimensional Hilbert space a linear equation, while the equation for > is a linear equation in infinite dimensions. More precisely, we have <\eqnarray*> Z>||-m|)>Z+V>|\Z>|)>+\,>>|Z>||-m|)>Z+\,>>>> where > is the functional derivative of\ <\equation*> V>=>exp|\QZ-|2>c>|> \z, with >> is the constant appearing in the definition of the Wick product appearing in the functional >>, and > and > are the projection of > on the spaces > and >>, respectively. Since the equation for > is a finite-dimensional equation with drift given by the gradient of a function and the equation for > is linear with self-adjoint drift, then the unique invariant measure of the process ,Z|)>> satisfies property , and hence also their sum +Z=X+Y> does. For further details on the relation between symmetric processes and the integration by parts formula see e.g.. <\remark> From the proof of the previous result it is evident that the system consists of two independent equations: an infinite-dimensional linear one and a finite-dimensional non-linear one. This means that >> is the unique invariant measure of the system. We prove tightness of the measures >>. <\theorem> The family of measures >|)>>> is tight in \B>>. <\proof> Let >, >, and > be the Lyapunov functions defined in equations,, and, respectively. Since the measures >> are solutions to the FPK equation associated with >> in the sense of Problem and \\>, then <\equation*> \>V \>X,\Y|)>=0. Moreover, we have by Theorem together with Remark and by Theorem together with Remark, that > is a Lyapunov function for >, where and solves equation and, respectively. Thus, by Lemma, we have <\equation*> V \>X,\Y|)>\V \>X,\Y|)>=V \X|)>. Since \>V \X|)>\+\>, and since > has compact sub-levels, the thesis follows. We get the following consequence. <\corollary> There exist three sequences |)>\>\\>, |)>\>\\>, and |)>\\> such that \+\>, \+\>, and \0> as +\>, respectively, and some probability measures ,\>>, >>, and > such that, for any > and any >, we have\ <\equation*> lim+\>\,M,\>=\,\>, for any > we have <\equation*> lim+\>\,\>=\>, and finally <\equation*> lim+\>\>=\. <\proof> The result follows from Theorem and a diagonalization argument. We want now to prove that any limit measure > built in Corollary solves the FPK equation in the sense of Problem. We first prove that any measure appearing in Corollary solves the FPK equation associated with the corresponding operator. For simplicity, let us drop the dependence on the parameters , , and introduced in Corollary. The aim of the section is to prove the following result. <\theorem> Let \\>. We have the limit as +\> and \0>, <\equation> \>\ \ \>\\\ \\. In order to prove the limit, we proceed by first taking +\>, then +\>, and eventually \0>, showing the convergence step by step. +\>> As mentioned above, we start off by taking +\>, namely we want to show that\ <\equation*> \>|)> \\>\\>|)> \\>,N\+\. The term in the operator >> involving only is independent of , in particular we do not need to consider the trace. Instead, we need to work on the terms involving only the derivatives with respect to. Since, by Corollary, >> is tight and converges weakly up to subsequences to some limit >>, we have to show that <\equation*> >|)>-\>|)>|]> \>X,\Y|)>\0,N\+\. Let us compute the integrand and rearrange its terms. We have <\eqnarray*> ||>|)>-\>|)>>>||||Q>\eQ>\X|)>>:eQ>\Y|)>>|)>|)>,\\|\>-\|g>\e>\X|)>>:e>\Y|)>>|)>,\\|\>>>|||+\\+\\\>>>> with <\equation> >|>||Q>\eQ>\X|)>>:eQ>\Y|)>>|)>|)>,\\|\>-\|g>\eQ>\X|)>>:eQ>\Y|)>>|)>,\\|\>>>||||g>\eQ>\X|)>>:eQ>\Y|)>>|)>,-I|)> Per>\\|\>>>>>> <\equation> \\\\|g>\eQ>\X|)>>:eQ>\Y|)>>|)>,\\|\>-\|g>\e>\X|)>>:eQ>\Y|)>>|)>,\\|\> and <\equation> \\\\\|g>\e>\X|)>>:eQ>\Y|)>>|)>,\\|\>-\|g>\e>\X|)>>:e>\Y|)>>|)>,\\|\> Let us deal with the term >. We need, for 1>, the bound <\equation> |\eQ>\X|)>>:eQ>\Y|)>>|\<\|\|\>>>> \\>\C>. Indeed, if estimate holds, then, by the regularization property of >> \U namely the continuity for any 0> of the operator >\|)>:L>|)>\B,\>|)>> \U and exploiting the fact that the norm |Q-I|\<\|\|\>>,\>,B>,\>|)>>> converges to zero as +\> for any 0> and \0>, we get the convergence of the term > to as +\>. Let us prove the bound. The exponential involving disappears, since 0>. Moreover, since \0>, we have that the Wick exponential can be written as the exponential divided by some positive constant >>,i.e. <\equation*> \eQ>\X|)>>:=C> eQ>\X|)>>, which converges to some finite number as +\>. Then, we have the inequality <\equation*> |\eQ>\X|)>>:|\<\|\|\>>>>\C> e|\Q>\X|)>|\<\|\|\>>>>>, and hence, by positivity of >, we obtain <\equation*> |\eQ>\X|)>>:|\<\|\|\>>>>\C>e|\>\X|)>|\<\|\|\>>>>>, which is in |)>> by Fernique's theorem (see Theorem2.8.5 in). We deal with the term \> given by. We have <\eqnarray*> \>|>||g>|\<\|\|\>>>|\eQ>\X|)>>:-\e>\X|)>>:|\<\|\|\>>>| eQ>\Y|)>>|\<\|\|\>>>>|Per>\\|\<\|\|\>>>,>>>> and, by stochastic estimates (see Proposition), we have convergence of the mean of the term \> to zero. For the term \\> given by we exploit the non-positivity of and replace the exponential by a bounded smooth function with bounded derivatives. Then we have that <\equation*> eQ>\Y|)>>\e>\Y|)>>,L>. provided <\equation*> Q>\Y|)>\g>\Y,L>. On the other hand, we have the bound <\equation*> |Q>\Y|)>-g>\Y|\<\|\|\>>>>\|Q-I|\<\|\|\>>>,L>|)>>|g>\Y|\<\|\|\>>>>. But |Q-I|\<\|\|\>>>,L>|)>>\0> depending only on , while |g>\Y|\<\|\|\>>>>\L> uniformly since by the proof of Theorem we have that the integral \ V \>X,\Y|)>> is bounded uniformly with respect to. +\>> We now have to take +\>, namely we want to show that\ <\equation> \>|)> \\>\\>|)> \\>,M\+\. Notice that the only -dependent term is now the trace-term |)>>>\\P>|)>>. By the same considerations as above, we have to show <\equation*> >|)>-\>|)>|]> \>X,\Y|)>\0,M\+\. Since \\> (see Definition and Remark), we have <\equation*> tr>\|\|>|)>\f>,tr>>\\|\|>|)>\f>,\\1, where f> \\\+\>. First, note that <\eqnarray*> > \\ P|)>-tr>\|)>>||>-I|)>\\ P|)>+tr>\ -I|)>|)>>>|||>-I|)>\\ P|)>+tr>\\>\> -I|)>|)>>>|||>\\>\> -I|)>|)>.>>>> Therefore, we have the bound <\equation*> > \\ P|)>-tr>\|)>|\|>\tr>\\>|\|>|)> |\>-I|)>|\<\|\|\>>|)>,L|)>|)>>. Now, let L|)>>, then we have <\equation*> >-I|)>|)>=\>\\\> >h-h|)>. Take \\\1>, then, considering the operator > norm, we have\ <\equation*> |\>-I|)>|\<\|\|\>>>\22\\\>>\> h \z+2\\>\> h \z. Since >=\>\-\>\\>-\|)>>>, we get <\eqnarray*> |||\>-I|)>|\<\|\|\>>>>>||>|-\|)>>2\\\>>\> h \z+2>\\>h \z.>>>> Now note\ <\equation*> 2\\\>>\> h \z\>>>|)>|)> h \z, and >>|)>\C>\\>\>>, to get <\eqnarray*> |\>-I|)>|\<\|\|\>>>>|>|-\|)>>C>>h \z+2>\\>h \z>>||>|-\|)>>>+1|)>>h \z.>>>> By taking the supremum over all with |h|\<\|\|\>>|)>>\1>, we then obtain\ <\equation*> |\>-I|)>|\<\|\|\>>|)>,L|)>|)>>\-\|)>>. Letting +\>, we get the limit. \0>> Now, we are left to show the convergence <\equation*> \>|)> \\>\\|)> \\,\\0. As above, we have to show that <\equation*> >|)>-\|)>|]> \>X,\Y|)>\0,\\0. Rewriting the integrand, we get <\equation> >|)>-\|)>>|||>\e>\X|)>>:e>\Y|)>>|)>|)>,\\|\>-\|\eX>:eY>,\\|\>>>|||+\\+\\\>>>>> with <\equation> \\\|>\e>\X|)>>:e>\Y|)>>|)>|)>,\\|\>-\|e>\X|)>>:e>\Y|)>>|)>,\\|\> <\equation> \\:=\|e>\X|)>>:e>\Y|)>>|)>,\\|\>-\|eX>:e>\Y|)>>|)>,\\|\> <\equation> \\\\\|\|eX>:e>\Y|)>>|)>,\\|\>-\|eX>:eY>|)>,\\|\> The stochastic estimates for exp>\X|)>|)>:> are done in Proposition. We now deal with the term >. We have the inequality <\eqnarray*> >|||\e>\X|)>>:e>\Y|)>>,>-I|)>\\\|\>>>||>||\e>\X|)>>:e>\Y|)>>|\<\|\|\>>>>|>-I|)>\\\|\<\|\|\>>>>>>||>||\e>\X|)>>:|\<\|\|\>>>> |e>\Y|)>>|\<\|\|\>>>> |g>-I|\<\|\|\>>>>,B>|)>> |\\|\<\|\|\>>>>>.>>>> Taking the expectation and exploiting the negativity of , we have <\eqnarray*> |||>\e>\X|)>>:e>\Y|)>>|)>|)>-\e>\X|)>>:e>\Y|)>>,\\|\>|]>>>||>||\e>\X|)>>:|\<\|\|\>>>>|]>|g>-I|\<\|\|\>>>>,B>|)>>|\\|\<\|\|\>>>\B,B>>|)>>>>||>||>-1|)>|\<\|\|\>>>>,B>|)>>,>>>> and this last term converges to zero as \0>. The convergence to zero of the term \> follows from Proposition. Finally for the term \\>, we proceed as follows. We have the bound <\equation> \\\\|\eX>:|\<\|\|\>>>>|Y|\<\|\|\>>>>>|g>-I|\<\|\|\>>>>,B>|)>>|)>\ 2|)>|\\|\<\|\|\>>>>>. Since >> is tight and |\expX|)>:|\<\|\|\>>>>> is uniformly integrable with respect to the measure>>, then, for any \0>, there exists a Borel subset >\B\B> such that <\equation*> >>|\eX>:|\<\|\|\>>>> \\>\\, and >>=sup>>|Y|\<\|\|\>>>>>\+\>. Therefore, we have <\eqnarray*> \\|\|> \\>>|>|>>|\eX>:>\Y|)>>-eY>|)>,\\|\>|\|> \\>+>>|\eX>:>\Y|)>>-eY>|)>,\\|\>|\|> \\>>>||>|>>|\\|\<\|\|\>>>>>|g>-I|\<\|\|\>>>>,B>|)>>>|\eX>:|\<\|\|\>>>> \\+2\|\\|\<\|\|\>>>>>.>>>> Thus, <\equation*> lim\0>\\|\|> \\>\2|\\|\<\|\|\>>>>>\, which gives convergence to zero by arbitrary choice of>. In order to show inequality, note that <\eqnarray*> \\|\|>>|>| |:expX|)>:>\Y|)>|)>-expY|)>|)>|\<\|\|\>>>,l,2\>> |\\|\<\|\|\>>>>>>>||>| |:expX|)>:>\Y|)>|)>-expY|)>|)>|\<\|\|\>>>> |\\|\<\|\|\>>>>>>>||>||:expX|)>:|\<\|\|\>>>>|>\Y|)>-Y|\<\|\|\>>>>|)>\|:expX|)>:|\<\|\|\>>>>|)>|]> |\\|\<\|\|\>>>>>>>||>||:expX|)>:|\<\|\|\>>>>+|:expX|)>:|\<\|\|\>>>>|)> |>\Y|)>-Y|\<\|\|\>>>>\2|)> |\\|\<\|\|\>>>>>,>>>> where |)>> and +1/l=1> such that \2-2/l>.Now, if > is small enough such that >>\B>, then we have the\ <\equation*> |>\Y|)>-Y|\<\|\|\>>>>\2|)>\|Y|\<\|\|\>>>>>|g>-I|\<\|\|\>>>>,B>|)>>|)>\ 2, which is converging to zero for every >>>\B>. We conclude the argument by Lebesgue dominated convergence theorem and with the reasonings similar as above. In this section, we collect some results about weighted Besov spaces. While we only focus on spaces defined on the whole space >, the results hold also for Besov spaces on the -dimensional torus>. Let us start by introducing Littlewood-Paley blocks. Let > and > be smooth non-negative functions from > into > satisfying the following properties: <\itemize-dot> |)>\B> and |)>\B\B>, ,\\1> and +0>\y|)>=1>, for any \>, |)>\supp\|)>|)>=\>, for 1>, \|)>|)>\supp\|)>|)>=\>, if \1>, where > denotes the ball centered at \> with radius 0>. We introduce the following notations: =\>, |)>=\\|)>>, =|^>>, and for any \|)>> we put =D\f>. Moreover, we write, for any \0>, >=|)>/2>>, and let >|)>> be the >-space with respect to the norm <\equation*> |f|\<\|\|\>>>>=>\>|)> \y|)>, where |]>>. <\definition> >>>Let \>, |]>>, and \\>. For \|)>>, we define the norm <\equation*> |f|\<\|\|\>>>>=|-1>2|\|\<\|\|\>>>>|>. The space >|)>> is the subset of |)>> such that the norm |\|\<\|\|\>>>>> is finite. In the case where >, the weighted Besov space ,\,\>|)>> is denoted by >|)>> and it is called with regularity. Moreover, if \\\>, the space >|)>> coincides with the Banach space of -Hölder-continuous functions.\ The relation between weighted Besov spaces is stated in the following result. <\proposition> Let ,p,q,q\|]>>, ,\\\>, and ,s\\> be such that ->\s->> and \\>. Then, we have the compact immersion <\equation*> B>,q,\>\B>,q,\>. <\proof> The proof can be found in Theorem6.7 in. If , the Besov space >> coincides with the Sobolev space >>, i.e.the space of measurable tempered distributions with bounded norm\ <\equation*> |f|\<\|\|\>>>>=>\> +1|)>f|)>\\y. The following theorem allows to extend products between a smooth function and a distribution to elements of Besov spaces. <\theorem> Let ,p,p,q,q,q\|]>>, ,\,\,s,s,s\\>, be such that\ <\equation> =>+>,=>+>,\=\+\,s+s\0,s=s\s. Consider the bilinear map :\|)>\\|)>\\|)>>, mapping \\=f\g>. Then, there exists a unique continuous extension of > as the map <\equation*> \:B>,q,\>|)>\B>,q,\>|)>\B>|)>, and we have, for any B>,q,\>|)>>, B>,q,\>|)>>,\ <\equation*> |\|\<\|\|\>>>>\|f|\<\|\|\>>>,q,\>>|g|\<\|\|\>>>,q,\>>. <\proof> See Section3.3 in for Besov spaces with exponential weights. The proof for polynomial weights follows in a similar way. In the case of products between a positive measure and an element of a Besov space, the previous result can be improved as follows. <\proposition> Consider the same parameters as in Theorem satisfying and \0>, \0>. Suppose that B>,q,\>|)>\L>|)>> and that \B>,q,\>|)>> is a positive measure, then we have <\equation*> |f\\|\<\|\|\>>>,q,\>>\|f|\<\|\|\>>>>\|\|\<\|\|\>>>,q,\>>. <\proof> See, e.g.,Lemma28 in. The next result is an interpolation estimate for Besov spaces. <\proposition> \ Consider ,p,q,q\|]>>, ,\\\> and ,s\\>, and write, for any \>,\ <\equation*> >>=|p>+|p>,>>=|q>+|q>,\>=\\+|)>\,s>=\s+|)>s. If B>,q,\>|)>\B>,q,\>|)>>, then B>>>,q>,\>>|)>>, and furthermore <\equation*> |f|\<\|\|\>>>>>,q>,\>>>\|f|\<\|\|\>>>,q,\>>>|f|\<\|\|\>>>,q,\>>>. <\proof> The proof is based on the fact that the complex interpolation of the two spaces >,q,\>|)>> and >,q,\>|)>> is given by>>>,q>,\>>|)>>. Such an interpolation is shown in Theorem6.4.5 in for unweighted Besov spaces. The proof for weighted spaces follows from the fact that B>> if and only if \>\B> (see Theorem6.5 in). We introduce now the heat kernel and present some of its properties. Let =e+m|)>t>>, we consider L>,B>|)>|)>> and define the on as <\equation*> e+m|)>>f=>P>f|)> \\. Notice that +m|)>>f> is a distribution.\ More precisely, if 0> and if >, \\\\>, is a measurable function then <\equation*> e+m|)>>f=>>|)>> e|4|)>>+m|)>>f,y|)> \y \\. We have the following regularization property for +m|)>>>. Let us remark that all the following results hold also in the case where L>,t|]>,B>|)>|)>>, where \t\t\+\>, and the operator +m|)>>f> is defined as the integral from > to ,t|]>>. <\theorem> Consider |]>>, |]>>, \>, and let L>,B>|)>|)>>. Then, for any ,\\0> such that +\\1>, we have\ <\equation> e+m|)>>f\B>>,B>>|)>|)>. Notice that states that we are gaining regularity > in time and > in space.\ In order to prove Theorem, we need the following result saying that when we apply the heat kernel at time we gain > in space-regularity, but we have to pay with a multiplicative factor of>>.\ <\lemma> Let 0> and consider B>|)>>. We have, for every 0>, <\equation*> |Pg|\<\|\|\>>>>|)>>\t>et>|g|\<\|\|\>>>>. <\proof> See Proposition5 in. We will also need the following lemma saying that giving up some space-regularity we can gain a factor >> on the right-hand side. <\lemma> Consider \\1> and B>|)>>. Then, for any 0>,\ <\equation*> ||)>g|\<\|\|\>>>>>\t>|g|\<\|\|\>>>>. <\proof> See Proposition6 in. <\render-proof|Proof of Theorem> We give the proof for unweighted Besov spaces, the general case follows the same lines. We use the difference characterization of space-time Besov spaces (see e.g.Theorem2.36 in or Chapter2.6.1 in), which yields <\eqnarray*> |e-m|)>>f|\<\|\|\>>>,B>|)>|)>>>|>||e-m|)>>f|\<\|\|\>>,B>|)>|)>>>>|||>t|\|>\1>|e-m|)>>ft|)>-e-m|)>>f|\<\|\|\>>>|t|\|>>> \t|)> \t.>>>> First, we prove that the first term on the right-hand side is finite. Write =e+m|)>>f>, we have <\eqnarray*> ||\<\|\|\>>,B>|)>|)>>>||>||\<\|\|\>>>> \t>>|||>|-1>|>Pf \k||-1>>>>\t>>||>|>>e>|-1>|e>f||-1>>>> \k|)>\t>>>> Lemma and Young inequality yield <\eqnarray*> ||\<\|\|\>>,B>|)>|)>>>|>|>>|]>>|>> e>|-1>|f||-1>>> \k|)>\t>>||>|>|]>>|>> e> \k|)>+|-1>|f||-1>>,B|)>|)>>,>>>> where the first integral on the last step is finite if and only if \1>.\ Consider now the difference term. We have\ <\eqnarray*> t|)>->||t>Pt-k>f \k++m|)>\t>|)>>Pf \k>>||>|+I.>>>> Now, by Lemma and Young inequality, <\eqnarray*> |||I|\<\|\|\>>>>|)>|\<\|\|\>>>>|>|>>t,0|]>>|t-k|)>>>|f|\<\|\|\>>>|)>>>||>|t|)>>|f|\<\|\|\>>,B|)>|)>>>>>> Consider \|~>\1-\>, then, by Lemma, <\eqnarray*> ||I|\<\|\|\>>>>|)>|\<\|\|\>>>>|>|t|)>|~>>|e+m|)>>f|\<\|\|\>>,B+2|~>>|)>>>>||>|t|)>|~>>|f|\<\|\|\>>,B|)>>,>>>> where we used the first part of the proof in the last step and the fact that +|~>\1>. Putting everything together, we get\ <\eqnarray*> >t|\|>\1>|t|)>-|\<\|\|\>>>|t|\|>>> \t|)> \t>|>||f|\<\|\|\>>,B|)>> t|\|>\1>t|)>|)>r>+t|)>|~>r>|t|\|>>> \t|)>>>||>||f|\<\|\|\>>,B|)>> t|\|>\1>t|\|>|~>-\|)>r>> \t|)>>>||>||f|\<\|\|\>>,B|)>>,>>>> which gives the result. Thanks to the heat kernel, we have another representation for weighted Besov spaces. <\proposition> Let \>, |]>>, and \> be such that <\equation*> k\. Consider a smooth and compactly supported function>. Then, for any 0>, we have the following equivalence between norms <\equation> |f|\<\|\|\>>>>\|\\|)>|\<\|\|\>>>>+>t|)>q>|\>f|)>|\<\|\|\>>>> t|t>|)>. <\proof> See Section2.6.4 in for a version of the theorem with a mass-less heat kernel. In particular, such a result differs from the one presented above since the integral with respect to in equation goes from to instead of going from zero to >. This extension is possible thanks to the regularization properties of the heat kernel and the exponential decay thereof, due to the presence of the positive mass0>. <\remark> Under the same hypotheses as in Proposition, if we assume further that 0>, then the norm |\\|)>|\<\|\|\>>>>> appearing in equation can be substituted by the weighted >-norm of, since we have\ <\equation*> |\\|)>|\<\|\|\>>>>\|f|\<\|\|\>>>>\|f|\<\|\|\>>>>. Namely, for 0>, equivalence becomes <\equation*> |f|\<\|\|\>>>>\|f|\<\|\|\>>>>+t|)>q>|\>f|)>|\<\|\|\>>>> t|t>|)>. We prove here some stochastic estimates for the term eg>\X>:>. We need two different kind of estimates. One has to be at the initial time with respect to the Gaussian free field, while the second one needs to estimate the term in some >-space with respect to the variable. Given >> as in Section and 0>, and > being the Gaussian free field with mass , we define the Wick exponential expg>\X|)>:> as in equation. The previous expression coincides with the standard Wick exponential in the case where we equip the space > with the free field measure>. <\proposition> Let \8\>, \0>. Then, for every 1> such that r/|)>\2>, and for every \0> such that \2>, and for every \0>, we have that the sequence <\equation*> \expg>\X|)>:\\expX|)>:,L,\|)>,B-\>>|)>|)>, where expX|)>:> is the unique (positive) limit random distribution and =\/|)>>. <\proof> We report here the proof for the case \4\>, the general case can be obtained mixing the method presented here and the techniques by. We consider first the case . We have, for =\|)>>, <\equation*> \>e>\X|)>>:-\eX>:|)>\K|]>|\|>>|)>\z|]>=>\e>\X|)>>:-\eX>:|)>\K|]>|\|>|]>>|)>\z. By translation invariance and orthogonality of Wick polynomials, we can consider <\eqnarray*> e>\X|)>>:-\eX>:|)>\K|]>|\|>|]>>|||e>\X|)>>:-\eX>:|)>,K|\>|\|>|]>>>|||>|> \|\>\X|)>:-\X:,K|\>|\|>|]>.>>>> It then suffices to show that, for each \>,\ <\equation> \|\>\X|)>:-\X:,K|\>|\|>|]>\0. Indeed, for |\|>\>>, we have\ <\equation*> >|> \|\>\X|)>:-\X:,K|\>|\|>|]>\2>|> \|\>\X|)>:,K|\>|\|>+|\X:,K|\>|\|>|]>. We have then to show <\equation*> |> \|\>\X|)>:,K|\>|\|>+|\:,K|\>|\|>|]>|\|>\c, for some |}>\\|)>>.\ We want an uniform bound on |\ \>\X|)>:|\<\|\|\>>>>, which would then give, as \0>, the convergence \>\X|)>:\\ \X:> almost surely.\ Notice that <\equation*> |\\>\X|)>:|\<\|\|\>>>=n!>KK|)>>|)>|)> \x \x, where <\equation*> \>=g>2>\\=\>|\|>||\|>+m>|)>. By Lemma2.2 in, we have, for some constant 0>, <\eqnarray*> >>|>|>log \\|)>+C,>>|>|>|>log +C.>>>> Moreover, if 0>, we have the point-wise limits <\equation*> \>|\\0>\. Now, <\eqnarray*> |\ \>\X|)>:|\<\|\|\>>>>||>KK|)>>|)>|)> \x \x>>||>|> log|\|>|)>,>>||\ \X:|\<\|\|\>>>>||>KK|)>|)>|)> \x \x>>||>|> log |\|>|)>,>>>> where the multiplicative constant absorbed in the symbol > does not depend on , >, and. Summing up, we have <\equation*> |>\|\|)>:,K|\>|\|>+|\|)>:,K|\>|\|>|]>|\|>\|n!>> log |\|>|)>|\|>\c. Then, <\equation*> c=c\|n!>> log |\|>|)>|\|>\xp|2\> log |\|>|)>\ 2/|)>>, which yields <\equation*> -1>2c\-1> 2/|)>+s|)>>. Therefore, we need\ <\equation*> s\-|4\>. Now, we have <\eqnarray*> |\exp>\X|)>|)>:-\expX|)>:|\<\|\|\>>>>|]>>||-1>2 \|\ \exp>\X|)>|)>:-\ \expX|)>:|\<\|\|\>>>>>||>|-1>2 \|K,\exp>\X|)>|)>:-\expX|)>:|\>|]>>>||>|-1>2 |> \|K,\>\X|)>:-\X:|\>|]>.>>>> Notice that each term in the sum converges to in and. Moreover, we have an uniform bound in and, since <\equation*> 2 |> \|K,\>\X|)>:-\X:|\>|]>\2 c, where the term on the right-hand side is summable, and therefore it is in|)>>. By Lebesgue dominated convergence theorem, everything converges to zero. We follow the proof of Theorem3.8 in to address the case 2>. Let 2>, take \1>, and recall hypercontractivity of Gaussian Wick monomials (see e.g.ChapterIII in): <\equation*> \X:|)>|]>\X:|)>|]>|)>. We have <\eqnarray*> ||e>\X|)>>:-\eX>:|)>\K|]>|\|>|]>>>||||e>\X|)>>:-\eX>:|)>,K|\>|\|>|]>>>|||>|n!> \|\>\X|)>:-\X:,K|\>|\|>|]>>>||>|>|)>|n!> \|\>\X|)>:-\X:,K|\>|\|>|]>>>||>||\-1>|)>>\|> \|\>\X|)>:-\X:,K|\>|\|>|]>|)>>>||>||\-1>|)> \|\e\>\X|)>>:-\e\X>:,K|\>|\|>|]>.>>>> Taking > such that <\equation*> s\-\|4\>, we conclude the proof in the same way as in the case . We need also a result for the periodic setting. In particular, we prove the convergence of the Wick exponential eQ>\X|)>>:> introduced in equation, as +\>..\ <\proposition> Recall that =\/|)>>. If p\2/\>, \0>, \0>, and \\>, for a positive constant depending on , then we have the convergence, as +\>, <\equation*> eQ>\X|)>>:-\e>\X|)>>:|)>\0,L>|)>,\|)>,L|)>|)>. <\proof> Notice that, if is a Gaussian random variable, then <\equation*> \eZ>:=eZ-|2>\|]>>. Therefore, applying the previous reasoning for >\X|)>> and first =\>, and then =p\>, then we have <\equation*> Q>\X|)>>:|)>=e p Q>\X|)>-p\\>\X|)>|)>|]>>=:ep Q>\X|)>>:e|2>-p|)>\>\X|)>|)>|]>>. We observe that, if C>|)>>, then, for every \>, <\equation*> Q>\X|)>\g>\X,N\+\. Moreover, we have that\ <\equation*> \>\X|)>|)>|]>\\>\X|)>|]>, and also\ <\equation*> :eQg>\X>:\L> , These last properties imply that\ <\equation> :eQ>\X|)>>:\:e>\X|)>>:,L>. Taking the norms, we have by translation invariance <\equation*> \>Q>\X|)>>:-:e>\X|)>>:|)>|\|>\z|]>=M|)>\Q>\X|)>>:-:e>\X|)>>:|)>|\|>|]>. and the last term converges to zero by. We now prove convergence in space-time of the Wick exponential.\ <\proposition> Consider the same parameters ,p,\,\> as in Proposition. Let |)>\>> be the solution of equation with > distributed as a Gaussian free field of mass. Then we have\ <\equation*> \expg>\X|)>:\\expX|)>:,L\\,L>,B-\>>|)>|)>|)>, where \0> is such that p\1>. <\proof> The proof follows closely the one of Proposition. See also Theorem3.2 in and Lemma2.5 in. Consider the partial differential equation\ <\equation> -\+m|)>\=-B|)>+C|)>,\\\\, where \\\H|)>\\> is a positive function with compact support with respect to \> independently of\>, >|)>\C>|)>> is a linear and bounded operator which is self-adjoint with respect to the |)>> Hilbert space structure and which commutes with the Laplacian >, while \\\\> is a measurable function such that, for any 0>, we have\ <\equation*> |B,\|)>|)>C|)>|\<\|\|\>>> \s\+\, where \L,H|)>|)>\L>,L|)>|)>>. In this section, we prove some a priori estimates for equation. <\theorem> Consider equation with \0>. We have the bound, for some constant 0> and for every \\1>, <\equation*> |A\|)>|\<\|\|\>>>+|A\|)>|\<\|\|\>>>+-K\|)>|A\|)>|\<\|\|\>>>|)> \s\|B,\|)>|)>C|)>|\<\|\|\>>> \s, where the constant implied in the symbol > does not depend neither on nor on . <\proof> Multiplying equation by > and integrating we have,\ <\eqnarray*> |||A\|)>|\<\|\|\>>>+|A\|)>|\<\|\|\>>>+m|A\|)>|\<\|\|\>>>|)> \s>>|||B|)>+C|)>A\ \z \s>>|||B|)>|)> \z \s>>|||B|)>CA\ \z \s>>||>|B|)>CA\ \z \s.>>>> Then, exploiting Young's inequality, <\eqnarray*> |||A\|)>|\<\|\|\>>>+|A\|)>|\<\|\|\>>>+m|A\|)>|\<\|\|\>>>|)> \s>>||>|B|)>CA\ \z \s>>||>|>|B|)>C|> \z \s+C\|A\|)>|\<\|\|\>>> \ \s.>>>> Since |A\|)>|\<\|\|\>>>=|AA\|)>|\<\|\|\>>>\|A\|)>|\<\|\|\>>>>, we can reabsorb the last term on the right-hand side and apply Young's convolution inequality to the remaining term to get <\eqnarray*> |||A\|)>|\<\|\|\>>>+|A\|)>|\<\|\|\>>>+-C\|)>|A\|)>|\<\|\|\>>>|)> \s>>||>|>|B|)>C|> \z \s>>||>||B,\|)>|)>C|)>|\<\|\|\>>> \s.>>>> This concludes the proof. Let us modify the previous result in order to deal with weighted norms. In particular, we consider the case where =g>\\>, where >> is defined as in Section, with \=>\\>. Let us also recall the definition \ >=|)>/2>>, for \> and 0>. <\theorem> Consider equation with \0>. Assume further that <\equation*> |B,\,\|)>|\<\|\|\>>>,L>>|)>|)>>\+\. We have the bound, for some constant ,C\0> and for every \,|~>\1>, <\eqnarray*> |||A\|)>|\<\|\|\>>/2>>+C|)>|A\|)>|\<\|\|\>>/2>>+-|~>C|)>|A\|)>|\<\|\|\>>/2>>|)> \s>>||>||B,\|)>|)> C|)>|\<\|\|\>>/2>> \s.>>>> <\proof> Instead of multiplying equation by >, we multiply it by >A\|)>>. Proceeding as in the previous proof and noticing that <\equation*> >A\|)>|\|>\\>\|\|>, \ we have, integrating by parts, <\eqnarray*> |||\/2>A\|)>|\<\|\|\>>>+|\/2>A\|)>|\<\|\|\>>>+m|\/2>A\|)>|\<\|\|\>>>|)> \s>>|||A\|)>\|)>\\> \z \s>>|||B|)>+C|)>A>A\|)> \z \s>>|||B|)>|)>> A\|)>|)> \z \s>>|||B|)>C> A\|)>|)> \z \s.>>>> Let us focus on the term\ <\equation*> A\|)> \|)> \\> \z \s. Multiplying and dividing by >> inside the integrals gives <\equation*> \> A\|)> \|)> \>|\>> \z \s. By Young's inequality we have <\eqnarray*> ||\> A\|)> \|)> \>|\>> \z \s|\|>>>||>|>\> \|)> \>|\>>|)> \z \s>>|||C\>A\|)> \z \s,>>>> and now we have <\equation*> \>|\>>\k|\|>|1+k>\\ sup\>|\|>|1+>\ C. This yields <\eqnarray*> |||\/2>A\|)>|\<\|\|\>>>+C|)>|\/2>A\|)>|\<\|\|\>>>+|m-C>|>|\/2>A\|)>|\<\|\|\>>>|)> \s>>||>|B|)>|)>> A\|)>|)> \z \s>>|||B|)>C> A\|)>|)> \z \s.>>>> Applying Young's inequality we get <\eqnarray*> |||\/2>A\|)>|\<\|\|\>>>+C|)>|\/2>A\|)>|\<\|\|\>>>+|m-C>|>|\/2>A\|)>|\<\|\|\>>>|)>\\s>>||>|>\> |)>|\|> |\|> \z \s+\\> \|\|> \z \s>>|||>/2>B|)>C|)> \z \s+\/2> A\|)> \z \s>>||>|>|B,\,\|)>|\<\|\|\>>>,L>>|)>|)>> |)>|\|> |\|> \z \s>>||||\/2>A\|)>|\<\|\|\>>> \s>>|||>|\/2> B,\|)>|)> C|)>|\<\|\|\>>> \s+\|\/2> A\|)>|\<\|\|\>>> \s,>>>> reabsorbing the terms multiplied by > and > respectively, to the left-hand side and noticing that from the proof of Theorem we also get <\equation*> B|)>|)> \z \s\|B,\|)>|)>C|)>|\<\|\|\>>> \s, we have, renaming the constants and introducing |~>>, <\eqnarray*> |||A\|)>|\<\|\|\>>/2>>+C|)>|A\|)>|\<\|\|\>>/2>>+-|~>C|)>|A\|)>|\<\|\|\>>/2>>|)> \s>>||>||B,\|)>|)> C|)>|\<\|\|\>>/2>> \s.>>>> This concludes the proof. We now apply a bootstrap argument to the previous result to get the following statement. <\corollary> Under the same hypotheses of Theorem, we have <\equation*> |\|\<\|\|\>>>>,B-\>>|)>|)>>\\|B|\<\|\|\>>>>,L>>|)>|)>>,|C|\<\|\|\>>>>,L>>|)>|)>>|)>, where > is a second degree polynomial. <\proof> From Theorem, we know that \> lives in >,H>|)>|)>\L>>,L>|)>|)>>. By interpolation and using the fact that >|)>\B>|)>>, for every +\>, we have\ <\equation*> |A\|\<\|\|\>>/2>,B/2>|)>|)>>\>\>|B,\|)>|)> C|)>|\<\|\|\>>/2>> \s. Applying the heat kernel to the equation, we have\ <\equation*> \=P|B|)>+C|)>|>\s, and therefore, if and \0>, we have by Theorem, <\equation*> |\|\<\|\|\>>>>,B-\>>|)>|)>>\|B,\,\,\|)>|)>,\|)>+C,\|)>|)>|\<\|\|\>>>,B>|)>|)>>. On the other hand, for some second degree polynomial >, we have <\eqnarray*> |||B,\,\,\|)>|)>,\|)>+C,\|)>|)>|\<\|\|\>>>,B>|)>|)>>>>||>||B|\<\|\|\>>>>,L>>|)>|)>>,|C|\<\|\|\>>>>,L>>|)>|)>>|)>.>>>> This concludes the proof. > We give here the proof of Lemma. We only show the first part of the result, the second one following in a straightforward way. It is sufficient to prove the following property: for any \> there exists a sequence of cylinder functions |)>\>> (here we consider a sequence |)>\>> depending on two parameters) such that G\\G> point-wise as +\>, and we have the uniform bound <\equation> G|\|>\F, for some measurable \L|)>>, e.g.some polynomial of. And secondly we have to show that >G\\G> point-wise as \0> with a bound >G|\|>\F>, for some \L|)>>. Since we take \\>>, the result follows from Lebesgue's dominated convergence theorem. Take \> and \>, and define <\equation*> G=GQX|)>,fQY|)>|)>, where =f/M|)>>, \> being a compactly supported smooth function in > which is identically in a neighborhood of the origin, and > is the operator defined in Section. We need to show that <\eqnarray*> G>|>|G,B>>|)>,>>|G>|>|G,B\|)>+\>>|)>l\|)>,>>|G|)>>|>|G|)>,.>>>> We focus only on the proof of and since the limit follows with similar arguments. We have, <\eqnarray*> ||G>>|||GQX|)>,fQY|)>|)>Qh|)>|]>,>>>> On the other hand, since the integrability parameters in the Besov spaces >, >, >>|)>>, and \|)>+\>>|)>> are finite, the linear operator fQZ|)>>, where \>, where > is anyone of the Besov spaces listed before, strongly converges to the identity >> on>. Therefore, by continuity of G>, we get that the following convergence holds as +\>: <\equation*> \GQX|)>,fQY|)>|)>\\G,B>>|)>. By strong convergence of the operator fQZ|)>> in >>|)>> and the fact that the composition of strongly convergent operators is strongly convergent, we get the limit in. Let us now prove. We have, with the notation =QX|)>,fQY|)>|)>>, and >, <\eqnarray*> |||Qf\G|)> fQf|)>-trG|)>>>|||Qf\G|)> fQf|)>-trQf\GfQf|)>>>|||Qf\GfQf|)>-trG|)>.>>>> Let us deal first with the second term of equation, that is <\equation> trQf\GfQf|)>-trG|)>. We have <\eqnarray*> Qf\GfQf|)>>||GQf|)>|)>>>|||\>GQf|)>h,h|)>>>>|||\>Qf|)>h,\Gh|)>>>>|||\>Qf|)>h,\Gh|)>>>>|||\>\Qf|)>h,h|)>>,>>>> where we use that >, >, and G> are self-adjoint operators in |)>>, and |)>\>> is an orthonormal basis of |)>> of eigenvectors related to the eigenvalues |)>\>> of the operator G> (which exists being G> a compact operator). Since G> is also a trace-class operator, we have that |\|>|)>\>\\|)>>. Furthermore, we get <\equation*> Qf|)>h,h|)>|\|>\|\|>|Qf|)>|\<\|\|\>>,L|)>>\|\|>\>|Qf|)>|\<\|\|\>>,L|)>>|)>\|\|>. Since we have the strong convergences \id|)>>> as +\>, and \id|)>>> as +\>, and so |f|\<\|\|\>>,L|)>>,sup|Q|\<\|\|\>>,L|)>>\+\>, then we have <\equation*> Qf|)>\id|)>> Thus, we have Qf|)>h\h> in > as +\>, and therefore\ <\equation*> Qf|)>h,h|)>|}>\>\|}>\> Then the term converges to zero by Lebesgue's dominated convergence theorem. We now show that the first term in equation, i.e. <\equation> trQf\G|)> fQf|)>-trQf\GfQf|)>. converges to zero. Let :H>>|)>\L|)>> be the natural isomorphism between the two spaces, where > and > are the same parameter as in pointof Definition. The natural identification of |)>> with its dual, allows us to identify the dual map >> of > with the natural isomorphism between |)>> and >>|)>>. We can then write <\equation*> \G= \G >>|)>, and therefore, for two points \\B>, <\eqnarray*> ||G-\G|)>|\|>|)>>>||>| \G>-\G|)>>|\|>>|)>|)>>>|||\>\G>-\G|)>>|\|>>|)>h,h|)>>>>|||\>\G>-\G|)>>|\|>>|)>h,>|)>h|)>>>>||>||\\G>-\G|)>>|\<\|\|\>>,L|)>>\>|>|)>h|\<\|\|\>>>>>||>||\\G>-\G|)>>|\<\|\|\>>,L|)>> tr>>\L> \>|)>\\H>>>|)>,>>>> which is finite since pointin Definition holds and the operator >>\L> \>|)>\\H>>>> is trace class because \1> and \1> (see Remark). By continuity of the map G:\B\L>>|)>,H>>|)>|)>> and the fact that, by similar arguments as the one exploited to show, \p> as +\>, we have that G-\G|)>|\|>|)>\0> as +\>. The argument is then concluded by the following observation: if we take the absolute value of the difference in, then\ <\eqnarray*> ||QfG|)>-\G|)> fQf|)>|\|>>>||>|QfG|)>-\G|\|> fQf|)>>>||>|\>|fQf|\<\|\|\>> trG|)>-\G|\|> |)>,>>>> which converges to zero since \>|fQf|\<\|\|\>>> is finite. In order to get inequality, we notice that <\equation> |\G|\<\|\|\>>\|\G|\<\|\|\>>\F, where > plays the role of >> in Definition, as well as similar inequalities for G> and G|)>>, which is due to the fact that the norm of operator Qf> is uniformly bounded in \>. The convergence >G\\G> as \0> is proved in Section. We consider here the system of equations\U, and give a proof of Proposition. For notation simplicity, we will write instead of >> when no confusion occurs. Moreover, if not explicitly specified, all the appearing parameters are assumed to be taken as in Definition. We start with a result about existence and uniqueness of solutions to the system of equations\U, that is the first part of the statement in Proposition. <\proposition> For any \0>, if ,Y|)>\\\B>|}>>, then there exists a unique solution > to equations\U such that\ <\equation*> ,Y|)>\\\B>|}>,t\\. <\proof> The unique solution to equation is given by the explicit formula <\equation> X=PX+P\ \s (see, e.g., Theorem5.4 in). Showing existence of a solution to equation is equivalent to showing existencefor\ <\equation*> =Y-PY. We proceed by a Schaefer's fixed point argument (see Theorem4 in Section9.2.2 in) to get the result up to a fixed time 0>. Consider the map > givenby, for >, <\equation*> \,Y,X|)>=-P\ f> \e>\X|)>>: e>\|)>> e P>\Y|)>> \s. We need to show that >,Y,X|)>:A\>> is a continuous and bounded map, where and >> are the convex closed subsets, for \0> small enough, \0>, \\\,1|)>>, \0> large enough, consisting of non-negative functions, and such that <\equation*> A\C-\>,C-\-\>+\>|)>|)>,>\C>,C-\>>|)>|)>. Exploiting the compact embedding <\equation*> C-\>,C-\-\>+\>|)>|)>\C>,C-\>>|)>|)>, given by Besov embedding and Corollary3 in, we can then proceed applying Schaefer's fixed point theorem to get existence for every compact subset of > of the form |]>>, for some \0>. We have <\eqnarray*> |||\,Y,X|)>|\<\|\|\>>>,C-\>>|)>|)>>>>||>||\ f> \e>\X|)>>: e>\|)>> eP>\Y|)>>|\<\|\|\>>>,L>>|)>|)>|)>>>>||>||\ f> \e>\X|)>>: eP>\Y|)>>|\<\|\|\>>>,L>>|)>|)>|)>>\+\,>>>> by the regularization property of the convolution with>>.\ We have to prove continuity. For every ,\A>, we have <\eqnarray*> |||\,Y,X|)>-\,Y,X|)>|\<\|\|\>>>,C-\>>|)>|)>>>>||>||\ f> \e>\X|)>>: >\|)>>-e>\|)>>|)>eP>\Y|)>>|\<\|\|\>>>,L>>|)>|)>|)>>>>||>||e>\|)>>-e>\|)>>|\<\|\|\>>>,L>>|)>|)>|)>>>>||>||g>\-g>\|\<\|\|\>>>,L>>|)>|)>|)>>>>||>||g>|\<\|\|\>>|)>> |-|\<\|\|\>>>,L>>|)>|)>|)>>>>||>||-|\<\|\|\>>-\>,C-\-\>+\>|)>|)>>.>>>> We are left to show uniqueness. Take two solutions and > to equation. Notice that their difference is given by =-> and it satisfies <\equation*> -\+m|)>-|)>=-\ f> \e>\X|)>>:>\Y|)>>-e>\Y|)>>|)>. Introduce a positive function \\> such that \0> when -\>, and with |\|>\C h>, for some constant 0>. Multiplying the previous expression by -|)>> and integrating with respect to time and space, we have <\eqnarray*> ||>h-|)>-\+m|)>-|)> \z \t>>|||>h-|)> f> \e>\X|)>>:>\Y|)>>-e>\Y|)>>|)> \z \t.>>>> For the left-hand side, we get by integration by parts <\eqnarray*> ||>h-|)>-\+m|)>-|)> \z \t>>||||-|\<\|\|\>>>-h|-|\<\|\|\>>> \t+h|-|\<\|\|\>>>+m|-|\<\|\|\>>>|)> \t>>||>||-|\<\|\|\>>>-|\|>|-|\<\|\|\>>> \t+h|-|\<\|\|\>>>+m|-|\<\|\|\>>>|)> \t>>||>||-|\<\|\|\>>>+h|-|\<\|\|\>>>+-C|)>|-|\<\|\|\>>>|)> \t.>>>> Notice that the last line is positive. Therefore,\ <\eqnarray*> |||-|\<\|\|\>>>+h |-|\<\|\|\>>>+-C|)>|-|\<\|\|\>>>|)> \t>>||>|>h-|)> \e>\X|)>>:>\Y|)>>-e>\Y|)>>|)> \z \t\0,>>>> which yields uniqueness. <\remark> The growth with respect to of the norm of the solution to equation is polynomial. Indeed, recalling that in the proof of Proposition we have\ <\equation*> >=-P \ f> :e>\X|)>>: e>\|)>> eP>\Y|)>> \s, we get that the time growth of |~>> is determined by the growth with respect to of the term <\equation*> \ f> \e>\X|)>>: e>\|)>> eP>\Y|)>>. Now, e+m|)>s>>\Y|)>|)>> does not increase in time, while >\|)>|)>> is bounded since \0>, and hence the growth with respect to is determined only by the remaining term, that is the exponential> \exp>\X|)>|)>:>. In the next result, we prove continuity of the solutions to equations\U with respect to the initial data. <\lemma> For every \0>, the solution > to equations\U are continuous with respect to > in > and with respect to > in \B>>, respectively.\ <\proof> As far as is concerned, continuity follows from the linearity of its representation. Let us focus on. Consider a sequence |)>\>> converging to some limit > in \B>>. Then |)>\>> is bounded in \B>>, and, by the regularization properties of>>, we have that PY|)>> is uniformly bounded with respect to \>, and on the support of>>. From inequality, it follows that the solution >|)>> is bounded in the space >\C>|]>,C-\>>|)>|)>> and pre-compact in C-\>|]>,C-\-\>+\>|)>|)>>. Thus, there exists a subsequence > such that >|)>> converges to some> in.\ On the other hand, since >|)>> solves the fixed-point equation <\equation*> >|)>=\>|)>,Y>,X|)>, and since > is continuous with respect to all its variables (see the proof of Proposition), we have <\equation*> =\,Y,X|)>. By uniqueness of solutions to the fixed point equation, we have that |)>=>. Finally, since the solution to equation with initial data >> is givenby <\equation*> Y>|)>=>|)>+PY>, by the continuity of the heat kernel and of >, we get the thesis. In this section, we denoteby and the solutions to the system of equations\U and we study their derivatives with respect to the initial data > and >.\ We now show point in Proposition. We have that is differentiable and that its derivatives >X>, >X>, and >X> solve equations, , and, respectively. This is because we have the following explicit representation (cf.equation) of the solution <\equation*> X=PX+P\ \s, which is linear with respect to the initial data. We immediately get that >X\0> and>X\0>. We now focus on the derivatives >Y>, >Y>, and >Y> of , and show that they exist and satisfy equations, , and, respectively. Furthermore, they are all continuous functions with respect to > and>. <\proposition> For every \0>, we have that that the derivatives >Y>, >Y>, and>Y> of the solution to equation exist and satisfy equations,, and, respectively. <\proof> We only give the proof for >Y>, the other cases follows in a similar way. Consider the approximating equation, thatis <\equation> -\+m|)>Y=-\>=-\f> \e>\X|)>>: e>\Y|)>>,Y=Y, with \0>. We denote the solution to equation as>, in order to stress the initial conditionB\B>>. We have the integral representation of the solution to: <\equation*> Y=PZ-P \>,Y|)> \s. In order to compute the derivative with respect to the initial data, we need to perturbit. Therefore, taking \0> and B\B>>, and considering the difference, we have <\equation*> Y+\h|)>-Y|)>=Ph\-P>,Y+\h|)>|)>-\>,Y|)>|)>|)> \s. But <\eqnarray*> ||P>,Y+\h|)>|)>-\>,Y|)>|)>|)> \s>>|||P\\>|X,\Y+\h|)>+|)>Y|)>|>|g>\+\h|)>-Y|)>|)>|> \\ \s,>>>> since >> is differentiable in the direction >\+\h|)>-Y|)>|)>>. Define\ <\equation*> H,h|)>=\\>|X,\Y+\h|)>+|)>Y|)>|> \\, so that we can write\ <\equation*> Y+\h|)>-Y|)>=Ph\-PH,h|)>\>\+\h|)>-Y|)>|)>|)> \s. Let us write down an equation for <\equation*> D,h>Y|)>=+\h|)>-Y|)>|\>. Notice that <\equation*> D,h>Y|)>=Ph-PH,h|)>\>\D,h>Y|)>|)> \s, and therefore <\equation> \,h>Y|)>|)>=-+m|)>D,h>Y|)>-H,h|)>>\D,h>Y|)>|)>,D,h>Y|)>=h. Now consider, as in the proof of Proposition, <\equation*> =Y-PY. Then, we have <\equation*> D,h>|)>=+\h|)>-|)>|\>=D,h>Y|)>-Ph, and such a difference satisfies the following equation <\equation> -\+m|)>D,h>|)>=-H,h|)>>\D,h>|)>+P>\h|)>|)>,D,h>Y|)>=0. By Theorem, we have then the following bound, for some constant 0> and every \\1>: <\eqnarray*> |||>\D,h>|)>|\<\|\|\>>>+|>\D,h>|)>|\<\|\|\>>>+-K\|)>|>\D,h>|)>|\<\|\|\>>>|)> \s>>||>||H,h|)>P>\h|)>|\<\|\|\>>> \s.>>>> Consider <\equation*> >|)>=Y+\h|)>-Ph\=PY-P\>,Y+\h|)>|)> \s. Notice that >|)>> is negative and moreover it solves <\equation*> >|)>=PY-\P f> \e>\X|)>>:e>\>|)>|)>> e\|g|\>>\Ph|>> \s. Recall that >> is such that >\h\L>> and >> is compactly supported. Therefore, <\padded-center> |)>>\Y|)>|)>|)>,expg>\>|)>|)>\L>,>>|>|>|)>> exp\>\Ph|)>|)>\L>,>>|>|)>>.>>>>> We then have the uniform estimate on given by, for any 0> small enough and p\+\>, <\eqnarray*> |H,h|)>|\<\|\|\>>>>>|>||f> \e>\X|)>>:|\<\|\|\>>>> |e\>\>|)>|)>>e\\|g>\Ph|>>e|)>>\Y|)>|)>>|\<\|\|\>>>>|)>|)>>>>||>||f> \e>\X|)>>:|\<\|\|\>>>> |e\\>\Ph|)>>|\<\|\|\>>>>|)>|)>>>>||>||f> \e>\X|)>>:|\<\|\|\>>>> |e>\Ph|\|>>|\<\|\|\>>>>|)>|)>>,>>>> which is uniform both in > and>. This gives existence of a limit for ,h>Y|)>> as \0>.\ We are left to show that the limit satisfies equation. First, we have to prove that the following limit holds <\equation*> lim\0> H,h|)>=\f> \e>\X|)>>:e>\Y|)>|)>>, in some suitable space.\ By Lemma, we have that +\h|)>\Y|)>> as \0> in\B>>, and hence, thanks to the regularization properties of >>, >\Y+\h|)>> converges to >\Y|)>> uniformly on the support of>>. Thus, we have the weak convergence <\equation> H,h|)>\\f> \e>\X|)>>:e>\Y|)>|)>>,\\0, and, by the uniform estimates and the compact embedding >\B>>, with \s> (see Proposition), we get the strong convergence in the space>>.\ We then have a weak limit for ,h>Y|)>> as \0>. Thus, proceeding as in the proof of Lemma, thanks to the a priori estimate, to the uniform bound, and to the convergence, we get that the limit of ,h>Y|)>> as \0> is a solution to equation. We prove here some bounds on >Y|)>>, >Y|)>>, and on the trace of >Y|)>>. Let us recall that the sets > and > are defined as in. <\proposition> For every \>, \|)>>, ,\\1> and >, we have the estimate <\eqnarray*> |||\>Y|)>|\<\|\|\>>>>,C>-2\>|)>|)>\L>,|)>>>>||>>>|||1>|f> \e>\X|)>>:e P>\h|)>>||1>>>>,L>>|)>|)>>,|h|\<\|\|\>>>|>,>>>> where > is a second degree polynomial. <\proof> We use the same notation as in the proof of Proposition. Recall that\ <\equation*> D,h>Y|)>=D,h>|)>+Ph, and then, since h> is uniformly bounded in >, in order to prove the result, it suffices to give an estimate on ,h>|)>> in the space >>,C>-2\>|)>|)>>. Let us consider a time-weight >> and a space-weight >> defines as\ <\equation*> \=|)>,z\\,\. Then, by Theorem we have, for some \\1>, <\eqnarray*> ||>|\>>\D,h>|)>|)>|\<\|\|\>>>>>|||\>|\>>\D,h>|)>|)>|\<\|\|\>>>+-\C|)>|\>>\D,h>|)>|)>|\<\|\|\>>>|)> \s>>||>|>\>\>>\,h|)>e+m|)>s>>\h|)>|)>|)> \x \s.>>>> Moreover, applying Corollary yields <\eqnarray*> |||H,h|)>>\D,h>|)>+P>\h|)>|)>|\<\|\|\>>>,B>|)>|)>>>>||>||H|\<\|\|\>>>>,L>>|)>|)>>,|>\h|\<\|\|\>>>>|)>>|)>.>>>> Together with the previous estimate, this yields uniform bounds on the norm |D,h>|)>|\<\|\|\>>>>,B-\>>|)>|)>>>, and choosing > and accordingly we then deduce that we have ,h>|)>\C>>,C>>|)>|)>>, uniformly in>. Now, letting \0> yields the result. <\proposition> For every \>, \|)>>, ,\\1> and >, we have the estimate <\equation*> |\>Y|)>|\<\|\|\>>>>,C>-2\>|)>|)>>\>>P||f> \e>\X|)>>:e P>\h|)>>|\<\|\|\>>>>,L>>|)>|)>>|>, where > is a second degree polynomial. <\proof> The proof is similar to the one of Proposition, the only difference is that here the initial data of >Y> is zero and therefore we do not need to subtract it before doing the estimates in Theorem and in Corollary. We now deal with the trace term appearing in the definition of the operator>. <\proposition> For every ,\\0>, there exist ,\\0> such that <\equation*> |\>Y|)>|\<\|\|\>>>>,H>>|)>>\>\f>|)> \e>\X|)>|)>>:e>\Y|)>|)>> e-m|)>t>|\|>>|)> \z|)>. It follows that, whenever \1> and \1>, <\equation*> tr>Y|)>\>|\|>|)>\>\f>|)> \e>\X|)>|)>>:e>\Y|)>|)>> e-m|)>t>|\|>>|)> \z|)>. <\proof> We suppose that all the computations involving equations and make sense since a rigorous proof of this fact can be given in a similar way as in the proof of Proposition. Let >> be the Dirac delta distribution, and consider the map, for \B> and \>, <\equation*> T,t>:H>>|)>\H>>|)>,h\\>Y|)>>|)>. Hereafter, we drop the dependence on > in the derivatives of. Let us then evaluate equations and at |)>>, and |)>|)>>, respectively, toget <\eqnarray*> ||--m|)>|)>\>Y|)>|)>>>|||f>|)> \eg>\X|)>>:eg>\Y|)>>>\\|)>|)>+>\\>Y|)>|)>|)>|)>>>|||f>|)> \e>\X|)>|)>>:e>\Y|)>|)>>g>|)>+>\\>Y|)>|)>|)>|)>,>>>> and <\eqnarray*> ||--m|)>|)>\>Y|)>|)>>>|||f> \e>\X|)>|)>>:e>\Y|)>|)>>>\\>Y|)>|)>|)>>>|||f> \e>\X|)>|)>>:e>\Y|)>|)>> >\Ph|)>|)>>\P\|)>|)>+>\\>Y|)>|)>|)>|)>>>|||f> \e>\X|)>|)>>:e>\Y|)>|)>> >\\>Y|)>|)> >\P\|)>|)>+>\\>Y|)>|)>|)>|)>.>>>> Using similar methods as in the proofs of Proposition and Proposition, it is possible to prove that >Y|)>> and >Y|)>> are differentiable infinitely many times with respect to and their derivatives with respect to for any multi-index > satisfy <\equation*> \>>\\>Y|)>|)>|)>=g>-y|)> \>\>Y|)> \y. Furthermore, by equations and, we get <\eqnarray*> --m|)>|)>\>\>Y|)>|)>>||f>|)> \e>\X|)>|)>>:e>\Y|)>|)>>P\>g>|)>>>|||f>|)> \e>\X|)>|)>>:e>\Y|)>|)>>>\\>\>Y|)>|)>|)>,>>>> and <\eqnarray*> ||--m|)>|)>\>\>Y|)>|)>>>|||f>|)> \e>\X|)>|)>>:e>\Y|)>|)>>>>g>\\>Y|)>|)>|)>>>|||f>|)> \e>\X|)>|)>>:e>\Y|)>|)>> >\Ph|)>|)>>>g>\P\|)>|)>+>\\>\>Y|)>|)>|)>|)>>>|||f>|)> \e>\X|)>|)>>:e>\Y|)>|)>> >\\>Y|)>|)>>>g>\P\|)>|)>+>\\>\>Y|)>|)>|)>|)>.>>>> Exploiting Theorem applied to equation, we get\ <\eqnarray*> |||>\\>\>Y|)>|\<\|\|\>>>+|>\\>\>Y>|)>|\<\|\|\>>> \\>>||>|\f>|)> :e>\X|)>|)>>:e>\Y|)>|)>> \>g>|)>|\|> \z\F>.>>>> By a bootstrap argument, i.e.applying Corollary, we can conclude that\ <\equation*> |\>\>Y|)>|\<\|\|\>>>\\>F>q>|)>, where > is an -dependent polynomial, because of the relation of the heat kernel with the support of>>. By a similar method as above, we get the following estimate concerning the second derivative <\equation*> |\>\>Y|)>|\<\|\|\>>>\|h|\<\|\|\>>>>>F>>|)>, where > is another -dependent polynomial. Taking large enough, we have <\equation*> |\>\>Y|)>|\<\|\|\>>>>\|h|\<\|\|\>>>>> F> >|)>. This proves that the map is linear with respect to as a map from >, moreover, if \> and fixing > and , the norm with respect to is givenby <\eqnarray*> |\>Y>|)>|)>|\<\|\|\>>>z|)>>>|||\|>\s>\> >\>Y|)>|)>|)> \z>>||>||\|>\s>\> |\>\>Y|)>|)>|\<\|\|\>>>z|)>> \z>>||>||h|\<\|\|\>>>>>|\|>\s>\> >>|)>|\|> \z.>>>> We are left to show that the last integral is finite. Recall <\equation*> F>=\f>|)>\e>\X|)>|)>>:e>\Y|)>|)>> \>g>|)>|\|> \z. Since we have to show that the integral is finite for some polynomial, we multiply (and divide) the heat kernel by some weight >>, to get <\equation*> F> \>=\f>|)> :e>\X|)>|)>>:e>\Y|)>|)>> > P\>g>|)>|\|> \z. By inequality(6.2) in Section6 of, we have, for some \0>, <\equation*> \>\\>|)>|\|>|)>>. Therefore, <\eqnarray*> > P\>g>|)>|\|>>||>t> e|\|>|2t>-mt> \>g>-|)>\> \|\|>>>||>|>t> e|\|>|2t>-mt> >g>-|)>|\|> \> \>>||>|>t> e|\|>|2t>-mt> >g>-|)>|\|> \>-|)> +|\|>|)>> \.>>>> By the compact support of >>, we have <\eqnarray*> > P\>g>|)>|\|>>|>|,g,\,\>>t> e|\|>|2t>-mt> +|\|>|)>> \>>|||,g,\,\> et> \+W|\|>|)>>|]>,>>>> where is some two-dimensional Brownian motion. Thus, if we take > large enough, we get <\eqnarray*> > P\>g>|)>|\|>>|>|,\,g,\,\> et> |\|>>+\|\|>>|]>|)>>>||>|,\,g,\,\,\> e-m|)>t>|\|>>|)>.>>>> Then, <\equation*> F>\>\C,\,g,\,\,\>\f>|)> :e>\X|)>|)>>:e>\Y|)>|)>> e-m|)>t>|\|>>|)> \z, which is finite, since >> is compactly supported. <\bibliography*|bib|tm-alpha|exp|References> <\bib-list|86> Sergio Albeverio, VladimirI.BogachevMichael Röckner. On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. , 52(3):325\U362, 1999. 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barashkov2021elliptic ustunel1995introduction hoshino2020stochastic hoshino2021stochastic daprato2014stochastic evans2010partial simon1987compact triebel2006theory3 <\associate|toc> |math-font-series||1Introduction> |.>>>>|> |Plan of the paper |.>>>>|> > |Notation |.>>>>|> > |Acknowledgements |.>>>>|> > |math-font-series||2Formulation of the problem> |.>>>>|> |2.1Reformulation of Problem |->|-0.3em|>|0em||0em|>> |.>>>>|> > |2.2A priori deductions for exponential interaction |.>>>>|> > |2.3A description of the exponential measure |.>>>>|> > |math-font-series||3Uniqueness of solution> |.>>>>|> |3.1Proof of uniqueness of solutions to Problem |->|-0.3em|>|0em||0em|>> |.>>>>|> > |3.2Analysis of the resolvent equation |.>>>>|> > |3.3Proof of uniqueness of solutions to Problem |->|-0.3em|>|0em||0em|>> |.>>>>|> > |math-font-series||4Existence of solutions via Lyapunov functions> |.>>>>|> |4.1Lyapunov functions |.>>>>|> > |4.2Measure of the approximating problem |.>>>>|> > |4.3Tightness of the measure |.>>>>|> > |4.4Limit of the operator |.>>>>|> > |4.4.1Limit as |N\+\> |.>>>>|> > |4.4.2Limit as |M\+\> |.>>>>|> > |4.4.3Limit as |\\0> |.>>>>|> > |math-font-series||Appendix ABesov spaces and heat semigroup> |.>>>>|> |math-font-series||Appendix BStochastic estimates for the Wick exponential> |.>>>>|> |math-font-series||Appendix CEstimates on linearized PDEs> |.>>>>|> |math-font-series||Appendix DProof of Lemma |->|-0.3em|>|0em||0em|>>> |.>>>>|> |math-font-series||Appendix ETechnical results for the resolvent equation> |.>>>>|> |E.1Flow equations |.>>>>|> > |E.2Derivatives of the flow |.>>>>|> > |E.2.1Existence and equations |.>>>>|> > |E.2.2Properties of the flow derivatives |.>>>>|> > |math-font-series||References> |.>>>>|>