<\body> <\hide-preamble> <\with|ornament-shape|rounded|ornament-color|pastel brown> <\ornamented> > ||<\author-affiliation> Hausdorff Center for Mathematics &\ Institute for Applied Mathematics University of Bonn, Germany |<\author-email> albeverio@iam.uni-bonn.de ||> \; >|> December 2018>> We prove an explicit formula for the law in zero of the solution of a class of elliptic (nonlinear) SPDE in >. This formula is the simplest instance of , discovered in the physics literature by Parisi and Sourlas(1979), which links the law of an elliptic SPDE in dimension with a Gibbs measure in dimensions. This phenomenon is similar to the relation between an > dimensional parabolic SPDE and its > dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of procedure in the sense of Nelson(1966) and Parisi and Wu(1981). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein,LandauandPerez(1984). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our context the arguments are non-trivial and a non-supersymmetric, elementary proof seems only to be available in the linear, i.e., Gaussian case. \; ||> <\table-of-contents|toc> Introduction> |.>>>>|> The elliptic SPDE> |.>>>>|> Strong solutions |.>>>>|> > Weak solutions |.>>>>|> > Elliptic stochastic quantization> |.>>>>|> Dimensional reduction> |.>>>>|> Bounded potentials |.>>>>|> > Potentials satisfying Hypothesis >> and |.>>>>|> > Potentials satisfying only Hypothesis >> |.>>>>|> > Supersymmetry> |.>>>>|> The superspace |.>>>>|> > The superfield |.>>>>|> > The supersymmetry |.>>>>|> > Expectation of supersymmetric polynomials |.>>>>|> > Removal of the spatial cut-off> |.>>>>|> Transformations in abstract Wiener spaces> |.>>>>|> Fermionic fields > |.>>>>|> |.>>>>|> broadly refers to the idea of sampling a given probability distribution by solving a stochastic differential equation (SDE). This idea is both appealing practically and theoretically since simulating or solving an SDE is sometimes simpler than sampling or studying a given distribution. If, in finite dimensions, this boils down mostly to the idea of the Monte Carlo Markov chain method (which was actually invented before stochastic quantization), it is in infinite dimensions that the method starts to have a real theoretical appeal.\ It was Nelson and subsequently Parisi and Wu who advocated the constructive use of stochastic partial differential equations (SPDEs) to realize a given Gibbs measure for the use of Euclidean quantum field theory (QFT). Indeed the original (parabolic) stochastic quantization procedure of can be understood as the equivalence <\equation> \|)>|]>\F|)>e|)>>\\. Here belongs to a suitable space of real-valued test functions, \> is an heuristic \PLebesgue measure\Q on |)>>, while on the left hand side the random field > depends on \\\\> and is a stationary solution to the parabolic SPDE <\equation> \\+-\|)>\+V|)>=\, where > is a Gaussian white noise in >, \\> a generic local potential bounded from below, > a positive parameter, and > is the fixed time marginal of > which has a law independent of by stationarity and on the right hand side we have the formal expression for a measure on functions on > with weight factor given by <\equation> S|)>\>\|\|>+m|\|>+V|)>\x. \ Eq. can be made mathematically precise and rigorous by tools from the theory of Markov processes, SDE/SPDEs and Dirichlet forms, for example when , or when the equation is regularized appropriately and, in certain cases, for suitable renormalized versions of the SPDE when . Let us note for example that in the full space it is easier to make sense of equation than of the formal Gibbs measure on the right hand side of, see. \; In a slightly different context, and inspired by previous perturbative computations of Imry and Ma, and Young, Parisi and Sourlas considered the solutions of the elliptic SPDEs <\equation> -\|)>\+V|)>=\ in > where > is a Gaussian white noise on > and they discovered that its stationary solutions are similarly related to the same dimensional Gibbs measure. If we take \> then, they claimed that, for \Pnice\Q test functions (e.g. correlation functions) we have <\equation> \|)>|)>|]>\F|)>eS|)>>\. More precisely the law of the random field |)>\>>, obtained by looking at the trace of > on the hyperplane ,\,x|)>\\:x=x=0|}>\\>, should be equivalent to that of the Gibbs measure formally appearing on the right hand side of and corresponding to the action functional. Therefore one can interpret equation as an prescription in the same spirit of equation. \; When one can directly check that the formula is correct. Indeed in this case the unique stationary solution > to the elliptic SPDE is given by a Gaussian process with covariance <\equation*> \\|)>|]>=>|)>>|+|)>>k||)>>,x,x\\. Therefore for all \\> we have <\equation*> \\|)>|]>=>e|)>>>q|+m+|)>>k||)>> <\equation*> =>q|+1|)>>>|)>>|m+>k||)>>=>>|)>>|m+>k||)>> where we performed a rescaling of the integral in order to decouple the two integrations. The reader can easily check that the expression we obtained describes the covariance of the Gaussian random field formally corresponding to the right hand side of for . \; While this last argument is almost trivial, a more general justification outside the Gaussian setting is not so obvious. The equivalence was derived in at the theoretical physics level of rigor going through a representation of the left hand side via a supersymmetric quantum field theory (QFT) involving a pair of scalar fermion fields. This is one of the instances of the phenomenon which is conjectured in certain random systems where the randomness effectively decreases the dimension of the space where fluctuations take place. \ A crucial assumption is that the equation has a unique solution, which is already a non-trivial problem for general . Parisi and Sourlas observed that non-uniqueness can lead to a breaking of the supersymmetry, in which case the relation could fail. So, part of the task of clarifying the situation is to determine under which conditions relations in the spirit of could anyway be true.\ \; The dimensional reduction of the elliptic SPDEs seems less amenable to standard probabilistic arguments than its parabolic counterpart. Let us remark that from the point of view of theoretical physics it is possible to justify also dimensional reduction in the parabolic case using a supersymmetric argument much like in the elliptic setting. \; The only attempt we are aware of to a mathematically rigorous understanding of the relation is the work of Klein, Landau and Perez (see also the related work on the density of states of electronic systems with random potentials) which however do not fully prove equation but only the equivalence between the intermediate supersymmetric theory in dimensions and the Gibbs measure in dimensions. The reason for this limitation is that the problem of uniqueness of the elliptic SPDE seems to unnecessarily restrict the class of potentials for which can be established and Klein et al. decided to bypass a detailed analysis of the situation by starting directly with the supersymmetric formulation. Their rigorous argument requires a cut-off, both on large momenta in \Porthogonal\Q dimensions and on the space variable in dimensions in order to obtain a well defined, finite volume problem. This regularization breaks the supersymmetry which has to be recovered by adding a suitable correction term, spoiling the final result (see Theorem and Theorem below). A subtle gap in their published proof is pointed out, and closed, in Section. \; Let us remark that, in a different context, dimensional reduction has been proven and exploited in the remarkable work of Brydges and Imbrie on branched polymers and more recently by Helmuth. \; In the present work we complete the program of , in case, by proving relation linking the solution to the ellptic SPDE with the Gibbs measure with action and removing the finite volume cut-off in some cases. \; Fix and consider the two dimensional elliptic multidimensional SPDE\ <\equation> -\|)>\+f\V|)>= \x\\ where =,\,\|)>> takes values in >, ,\,\|)>> are independent Gaussian white noises, \\> a smooth potential function, \|)>> with :\\\> a decreasing cut-off function, such that the derivative > of the function > is defined, tending to at infinity, and V=V|)>,n>> denotes the gradient of . We will denote \|)>>. \; Eq. is the elliptic counterpart of the equilibrium Langevin reversible dynamics for finite dimensional Gibbs measures. Let us note that the elliptic dynamics is already described by an SPDE in two dimensions while in the parabolic setting one would consider a much simpler Markovian SDE (no renormalization being necessary). The question of uniqueness of solutions is however quite similar in difficulty, indeed it is non-trivial to establish uniqueness of stationary solutions to the SDE and much work in the theory of long time behavior of Markov processes is devoted precisely to this. In the elliptic context of there is no (easy) Markov property helping and the question of uniqueness of weak stationary solutions seems more open, even in the presence of the cut-off.\ \; What makes this problem very interesting, is above all the fact that while the statements we would like to prove are quite easy to describe (see below), to our surprise their rigorous justification is already quite involved and not yet quite complete in full generality.\ \; Define the probability measure > on > by <\equation> \|\ y>\Z>exp|2>+V|)>|]>, where \>, >\>exp|2>+V|)>|]>\ y> (>> is well defined since is bounded from below). \; The main result of this paper is the following theorem which states that on very general conditions on there is always a weak solution which satisfies (an approximate) elliptic stochastic quantization relation (of the form). By weak solution to the SPDE we mean a probability measure > on the space of fields > under which -\|)>\+\V|)>> is distributed like Gaussian white noise on >. A strong solution > to equation is a measurable map \\=\|)>> satisfying the equation for almost all realizations of >. In order to state precisely our results we need to introduce \ the following assumptions on and on the finite volume cut-off: <\description-paragraphs> C. (convexity)>The potential \\> is a positive smooth function such that <\equation*> y\\\V+m is strictly convex and with its first and second partial derivatives grow at most exponentially at infinity. QC. (quasi convexity)>The potential \\> is a positive smooth function, such that it and its first and second partial derivatives grow at most exponentially at infinity and moreover it is such that there exists a function \\> with exponential growth at infinity such that we have <\equation*> -|,\V|)>|\>\H,\\,y\\> and \>>, with > is the dimensional unit sphere. CO. (cut-off) >The function is real valued, has at least > smoothness and in addition satisfies \0>, it decays exponentially at infinity and fulfills \bf> for \4m> (some examples of such functions are given in and the motivations for this hypothesis are explained in Remark below). <\theorem> Under the HypothesesQC andCO there exists (at least) one weak solution |~>> to equation such that for all measurable bounded functions \\> we have <\equation> |~>>h|)>\|)> |~>\|)>=Z>h\\ where |)>\e>fV|)>\x>> and \>\|)>|~>\|)>>. |~>> is a suitable Banach space of functions from > to > where |~>> is defined (see Section equations,and for a precise definition of |~>> and |~>>). <\remark> The following families of functions satisfy HypothesisQC: <\itemize-dot> Smooth convex functions (since they satisfy the stronger HypothesisC), Smooth bounded functions, Smooth functions having the second derivative semidefinite positive outside a compact set, Any positive linear combinations of the previous functions. The drawback of this result is the lack of constructive determination of the weak solution > for which the dimensional reduction described by equation is realized. This is linked with the fact that HypothesisQC does not guarantee uniqueness of strong solutions to eq.. The fact that non-uniqueness is related to a possible breaking of the supersymmetry associated with was already noted by Parisi and Sourlas. If we are willing to assume that the potential is convex we can be more precise, as the following theorem shows. <\theorem> Under HypothesesC andCO there exists an unique strong solution =\|)>> of equation and for all measurable bounded functions \\> we have <\equation> \|)>\|)>|]>=Z>h\\ where > is defined as in Theorem , \\|)>|]>>, and > denotes expectation with respect to the law of >. Both theorems require the presence of a suitable cut-off 1> which is responsible for the weighting factor |)>> on the left hand side of the dimensional reduction statements and. If we would be allowed to take 1> then we would have proven the version of equation. However, presently we are not able to do this for all QC potentials but only for those satisfying HypothesisC (see Section for the proof). \ <\theorem> Suppose that satisfies HypothesisC and let > be the unique strong solution in >;\|)>> (see Section for the definition of this space) of the equation <\equation> -\|)>\+\V|)>=\. Then for any \> and any measurable and bounded function defined on > we have <\equation> \|)>|]>=>h\\. This result is the first rigorous result on elliptic stochastic quantization without any cut-off. In fact in this case the results of Klein, Landau and Perez do not hold, since they use only an integral representation of the solution to equation which has meaning only when 1>. <\remark> It is easy to generalize Theorems, and to equations of the form <\equation> -\|)>\+\\f\>V|)>=\ \, where is as before, =|)>,n>> is an n> invertible matrix and the HypothesisC and QC are generalized accordingly. The paper is organized as follows. In Section we introduce the notions of strong and weak solutions to equation, and we prove existence of strong solutions (and thus also of weak solutions) under HypothesisQC in Theorem. We also provide, in Theorem, a representation of weak solutions via the theory of transformation of measures on Wiener spaces developed by Üstünel and Zakai in (whose setting and main facts needed here are summarized in Appendix).\ Section is devoted to the proof Theorem and Theorem about elliptic stochastic quantization, under the HypothesisQC andCO and using Theorem and PDE techniques. In Section Theorem is proven, i.e. dimensional reduction using Hypothesis>> (see Section ). The proof of Theorem is similar to the rigorous version of Parisi and Sourlas result proposed in, starting from different hypotheses. The proof of in Section is based on Theorem stating a relation between the expectation involving some bosonic and fermionic free fields.\ In Section we prove Theorem exploiting the properties of a supersymmetric Gaussian fields. In Section we also propose a brief introduction to supersymmetry and supersymmetric Gaussian fields used in an essential in the proof of Theorem.\ Section discusses the proof of Theorem on the cut-off removal under Hypothesis C. Appendix is a brief introduction to the theory of transformations on abstract Wiener spaces used in this paper, and Appendix consists in a discussion of some properties of fermionic fields.\ \; The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Scaling limits, rough paths, quantum field theory when work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/R014604/1 and by the German Research Foundation (DFG) via CRC 1060. In order to study equation we have to recall some definitions, notations and conventions. Fix an abstract Wiener space ,\,\|)>> where the noise > is defined (for the concept of abstract Wiener space we refer e.g. to). The Cameron-Martin space > is the space <\equation*> \\L;\|)>, with its natural scalar product and natural norm given by h,g\=>hg\x >. Let > (in which > is densely embedded) be the space <\equation> \=\>\W>>;\|)>\|)>>;\|)>|)>, where 1>, \0> and >>;\|)>> is a fractional Sobolev space with norm <\equation*> \<\|\|\>g\<\|\|\>>>>\>|)>>|)>-\>|\|>\x|)>>, for some \0> small enough and |)>>;\|)>|)>> is the space of the second order distributional derivatives of continuous functions on > growing at infinity at most as >> with norm\ <\equation*> |g|\<\|\|\>>+1|)>>|)>>\||)>>|)>g|)>|\<\|\|\>>>>. Thus >> is a Banach space with norm given by the sum of the norms of >>;\|)>> and of |)>>;\|)>|)>>. In the following we usually do not specify the indices > and in the definition of >> and we write only >. We also introduce the notation\ <\equation> |~>=|)>|)> The Gaussian measure > on > is the standard Gaussian measure with Fourier transform given by |\|\<\|\|\>>>>>. The white noise > is then naturally realized on ,\,\|)>>, in the sense that > is the random variable :\\;\|)>> (where ;\|)>> is the space of >\Uvalued Schwartz distributions on >) defined as =id>=w>. In this way the law of > is simply > (or, better, it is equal to the pushforward of > on \,\|)>> with respect the natural inclusion map of > in >). Sometimes it is also useful to consider the space >>> of >-Hölder continuous functions such that they and their derivatives (or Hölder norms) grow at infinity at most like >> for a real number > (this notation is used also if > is negative in that case the functions decrease at least like >>>). It is important to note that >>> can be identified with the Besov space >,\,\>|)>> (where >,\,\>|)>> is the weighted Besov space >,\>,|)>>|)>> \ of,Chapter 2 Section 2.7). It is also important to realize that |)>|)>\>>> if we choose big enough and \0> small enough. \; We introduce now two notions of solutions for equation. For later convenience it is better to discuss the equation in term of the variable \-\|)>\> for which it reads <\equation> \+f\V\|)>=\+U|)>=\, where <\equation*> \\-\|)> and where we introduced the map \\> given by <\equation> U\f\Vw|)>,w\\. Under the condition of (at most) exponential growth at infinity of , required by HypothesisQC and HypothesisC, it is possible to prove, that for \1> in the definition of >, for each > we have \>. Indeed we have <\equation*> |U|\<\|\|\>>|)>|)>>\|>|\<\|\|\>>|)>>\|>Vw|)>|\|>|\<\|\|\>>> and |>Vw|)>|\|>|\<\|\|\>>>> is finite since decreases exponentially at infinity and grow at most exponentially at infinity. Furthermore we introduce the map \\> as <\equation*> T\w+U. It is clear that a map \\ > satisfies equation, i.e. |)>=\=w>, for (>-)almost all \>, if and only if S> satisfies equation . The law > on > associated to a solution of equation must satisfy the relation >|)>=\>. For these reasons we introduce the following definition. <\definition> A measurable map \\> is a to equation if S=Id>> >-almost surely. A probability measure \\|)>> (where |)>> is the space of probability measures on >) on the space > is a to equations if >|)>=\>, where >> is the pushforward related with the map . If > is a probability measure on the space >, we write |~>> the unique probability measure on |~>> such that\ <\equation> +m|)>>|)>=|~>. In order to study the existence of strong solutions to equation we introduce an equivalent version of the same equation that is simpler to study. Indeed if we write <\equation*> |\>=\-\ \, and we suppose that > satisfies equation , then the function |\>> satisfies the following (random) PDE\ <\equation> -\|)>|\>+f\V|\>-\\|)>=0. Equation can be studied pathwise for any realization of the random field \>. Hereafter the symbol > stands for inequality with some positive constant standing on the right hand side. <\lemma> Suppose that satisfies HypothesisQC, and let |\>> be a classical > solution to the equation, such that \>|\>=0>, then for any \\2> and \0> we have\ <\equation> |\>\<\|\|\>>>|1+\<\|\|\>f e\|\|>>\<\|\|\>>>>>>> <\equation> |\>\<\|\|\>>>>>|>|\<\|\|\>|\>\<\|\|\>>>\<\|\|\>+1|)>>f e\|\|>>\<\|\|\>>,>>>>> for some positive constant > and where it and the constants involved in the symbol > depend only on the function in HypothesisQC. <\proof> Putting |\>>=|\>|)>>=|\>|\|>>, \>, since the > function |\>> converges to zero at infinity, the function |\>>> must have a global maximum at some point >\\>. This means that |\>>|)>>|)>\0>. On the other hand since |\>> solves equation we have <\eqnarray> r|\>>>|)>>|>|\|\>>|)>>|)>+mr|\>>>|)>>>||>||\>\\|\>-|\>|\|>+m|\>|\|>|)>>>||>|>|)>r|\>>>|)>|\>>>|)>\\V\>|)>+|\>>>|)>r|\>>>|)>|)>|)>>>>> where |\>>=|\>||\>|\|>>\\> when |\>>0>, and elsewhere. Using HypothesisQC we obtain <\equation*> |r|\>>|\<\|\|\>>>\>|)>H\>|)>|)>|m>\1+\<\|\|\>f e\|\|>>\<\|\|\>>, since grows at most exponentially at infinity. This result implies inequality .\ The bound can be obtained directly using the fact |\|\<\|\|\>>>>\|+m|)>|)>|\<\|\|\>>>>, where we use the properties of the Besov spaces >|)>=B>,\>|)>> with respect to derivatives (see, Chapter2 Section2.3.8). <\remark> It is simple to prove that the inequalities and can be chosen to be uniform with respect to some rescaling of the potential of the form V>, or satisfying Hypothesis>> below, where \>. In the following we denote by :\\>;\|)>|)>> the set valued function which associates to a given \> the (possible empty) set of solutions to equation in >;\|)>>, where \0>, when \> is evaluated in .\ <\theorem> Let be a smooth positive function satisfying HypothesisQC, then for any \> the set > is non-empty and closed. Furthermore \C;\|)>> and if \> is a bounded set then =B>> is compact in >>;\|)>> for any \0> and \0>. <\proof> We introduce the map >>;\|)>\\\|\>,w|)>\|\>,w|)>\>;\|)>>, where \ \\>, given by\ <\equation*> |\>,w|)>\-\V|\>+\\|)>|)>. The map > is continuous with respect to its first argument, indeed if |\>,|\>\>;\|)>>, <\eqnarray*> ||\>,w|)>-|\>,w|)>\<\|\|\>>>>>|>|||+1|)>>fV|\>,\\|)>-\V|\>,\\|)>|)>|\<\|\|\>>>>|>|||+1|)>>f \V|\>-t|\>-|\>|)>+\\|)>\|\>-|\>|)>\ t|\<\|\|\>>>>|>||\<\|\|\>|\>-|\>\<\|\|\>>|+1|)>>|\<\|\|\>>>|\V|\<\|\|\>>>+e||\>-|\>|\<\|\|\>>>>|e\|\|>>|\<\|\|\>>>|)>,>|>>> where the positive constant > depends on the exponential growth of V> at infinity. By a similar reasoning we can prove that > sends bounded sets of >>> into bounded sets of >>>, where \\> and \\>. Since the immersion >>\>>> is compact we have that > is a compact map. Since \\>> and |\>\>>;\|)>> we have +m|)>|\>,w|)>\;\|)>>. This implies, using the regularity results for Poisson equations (see Theorem4.3 in) and a bootstrap argument, that if |\>=|\>,w|)>> then |\>\C|)>>. From this fact it follows that, using inequalities and of Lemma and Remark, the solutions to the equation |\>=\|\>,w|)>> are uniformly bounded for \>. Thanks to these properties of the map > we can use Schaefer's fixed-point theorem (see Theorem4 Section 9.2 Chapter 9) to prove the existence of at least one solution to equation. Finally using again Lemma we have that > is compact for any bounded set \>. <\theorem> Under HypothesisQC on there exists a strong solution to equation (or equivalently to equation). <\proof> For proving the existence of a strong solution to the equation (in the sense of Definition) it is sufficient to prove that we can choose the solutions to equation, whose existence for any \> is guaranteed by Theorem, in a measurable way with respect \>. More precisely we have to prove that there exists a measurable selection for the function set map >, i.e. there exists a map >:\\>>> such that >\\>. Fix a sequence of balls ,\,B,\\\> of increasing radius and such that +\>B=\>, then, by Theorem, the map |\|\|>\B>> takes values in a compact set. As proven in Theorem the map > is continuous in |\>> and measurable in and therefore a Carathéodory map. As a consequence, by Filippov's implicit function theorem(see Theorem18.17 in), there exists a (Borel) measurable function >> defined on \B> such that >\>. The map >> defined on \B> by |>|\|>\B>=>> is the measurable selection that we need (since \B> is measurable). A strong solution to equation is then given by \w+-\|)>>>, \>. \ <\corollary> Under the HypothesisC there exists only one strong solution to equation. <\proof> Suppose that ,S> are two strong solutions to equation then, letting =\|)>|)>>, , writing \=\-\> and r=\|)>>>, we obtain\ <\equation*> -\|)>r|)>+2\\|\|>|)>+f\r \>\V|)>-\V|)>|)>|]>=0. By Lagrange's theorem there exists a function >, \>, taking values in the segment ,\|]>\\> such that \>\V|)>-\V|)>|)>=\r\V\>,\>|)>>. From this fact we obtain <\equation*> -\|)>r|)>+fV\>,\>|)>|)>\r\0. Since +\V\>,\>|)>\\\0> , V+m> being strictly convex by our HypothesisC, and r> is positive and goes to zero as +\>, we have =\> and therefore =S>. First of all we prove that the map , given by , is a > function (in the sense of, see Appendix) for the abstract Wiener space ,\,\|)>>. <\proposition> If and its derivatives grow at most exponentially at infinity, then the map is a > function, on the abstract Wiener space ,\,\|)>> and we have <\equation*> \U=f\\>Vw|)>\\|)>. Furthermore is > Fréchet differentiable as a map from > into . <\proof> The proof is essentially based on the fundamental theorem of calculus and the use of the Fourier transform. In order to give an idea of the proof we only \ prove the most difficult part, namely that U> is continuous with respect to translations by elements of >, where continuity is understood with respect to the Hilbert-Schmidt norm for operators acting on >. For fixed \>, \\> we have, for ,n>:\ <\equation> <\multline*> U|)>-\U=>>|\\\>V-\|)>|)>|)>\\|)>\\|)>\ t,>>>> where the sum over ,n> is implied. We recall that the Hilbert-Schmidt norm of an integral kernel is the integral of the square of the absolute value of the kernel. In our case the Fourier transform of the integral kernel representing the difference is given by <\equation*> |)>=>|)>|-k|\|>+m|)>>\-k|)>|-k|\|>+m|)>>k\k||)>>, where > is the Fourier transform of \\>V|)>|)>>, >. It is simple to prove that\ <\equation*> \<\|\|\>\U|)>|]>-\U|]>\<\|\|\>\>|)>,k|)>|\|>\k \k\ <\equation*> \\<\|\|\>ew|\|>+\h|\|>>\<\|\|\>>\<\|\|\>\<\|\|\>>\<\|\|\>h\<\|\|\>>, where > depends on the exponential growth of V>. Since ew|\|>+\h|\|>>\<\|\|\>>> is always finite in > (for > positive and small enough) we have proved the continuity of the map \\U|)>> with respect to the Hilbert-Schmidt norm. By the notation >+K|)>> we denote the regularized Fredholm determinant (see Appendix and also, Chapter9) which is well defined when is a Hilbert-Schmidt operator. The function >+\|)>> is a smooth functional from the space of Hilbert-Schmidt operators (with its natural norm) to > (see Theorem9.2 for the proof of this fact). We define the measurable map \\\|}>> <\equation*> N\\> to the equation =w>>|)>, moreover let \> be the set of zeros of the continuous function \\det>+\U|)>>. <\theorem> The function is greater or equal to and it is >-almost surely finite. Furthermore the map is proper. <\proof> We define |^>,w|)>=|^>+U|^>+w|)>>. Obviously we have that is a solution to the equation =w> if and only if |^>=z-w> is a solution to the equation |^>,w|)>=0>. On the other hand |^>> is solution to the equation |^>,w|)>=0> if and only if |\>=\|^>|)>> is a solution to equation. By Theorem, equation has at least one solution for any \> and so \1> for any \>. Let be a compact set in > we have that \K+-\|)>|)>> (where > is the set valued map introduced in Theorem). Since is compact, by Theorem, > is compact in >> which implies that -\|)>|)>> is compact in >>. Since the immersion >\\> is compact and the sum of two compact sets is compact, we obtain that is a proper map. Since by Proposition, |)>=0>, for proving the theorem it is sufficient to prove that \+\> for >T>. If >T> then +|\U|\|>>> is a linear invertible operator on > and so >+\U> is a linear invertible operator on >. By the implicit function theorem, we have that is a > diffeomorphism between a neighborhood > of onto |)>>. This implies that the set > consists of isolated points. Since the map is proper, this means that > is a compact set made only by isolated points which implies that > is a finite set. \ If \> is an > function we denote by :\\> the well defined Skorokhod integral of the map (see Appendix for an informal introduction of the concept, AppendixB of for a more detailed treatment and Proposition3.4.1 of for the proof of the fact that the Skorokhod integral of an > function is well defined). <\theorem> A probability measure > is a weak solution to equation if and only if it is absolutely continuous with respect to > and there exists a non-negative function L>|)>> such that T>A=1> for >-almost all \> and \|\\>=A|\|>> with <\equation*> \\detU|)>exp-\<\|\|\>U\<\|\|\>>|)>. <\proof> Recall that, by Proposition, |)>=0>. This implies that for any weak solution > we have |)>|)>=0>. Letting \T\T|)>> we deduce that \|)>=\|)>=1> and if we prove that > is absolutely continuous with respect to > on each > we have proved that > is absolutely continuous with respect to >. Using times iteratively the Kuratowski-Ryll-Nardzewski selection theorem(see Theorem18.13 in) due to the fact that \\> is composed by zero or elements, we can decompose the set > into measurable subsets ,\,\> where the map |T|\|>>> is invertible. This means that if \\> we have \\|)>\\|)>|)>>. On the other hand we have that |)>|)>=\\>|\|>\\>. This implies that if |)>=0> then \\|)>\\|)>|)>=\\>|\|>\\=0>. As a consequence |)>=\\\|)>=0> and > is absolutely continuous with respect to >. Theorem below implies that for any measurable positive functions we have <\equation> f\TA|\|>\\=fT>A|)>\ \. Taking >> and we deduce that |)>>|\|>\\=\|)>=0>. Therefore we can suppose that there exists a specific non-negative function such that \=A|\|>\\> and since >|)>=\> we must have <\equation*> f\\=f\T\\=f\TA|\|>\\, for any bounded measurable function . Comparing this with we deduce that T>A=1> for (>-)almost all \.>\ On the other hand, using again Theorem it is simple to prove that if \=A|\|>\\> and T>A=1> then > is a weak solution to equation. <\remark> If is any strong solution to equation then =S>\> is a weak solution. Furthermore it is simple to prove that the weak solutions of the form >\>, where is some strong solution to , are the extremes of the convex set \> satisfying >\=\>>|}>>. Using a lemma (precisely Lemma) that we shall prove below, it follows from this that > is weakly compact and thus, by Krein\UMilman theorem (see Theorem3.21 in), any measure \\> can be written as convex combination of measures induced by strong solutions. \ <\corollary> If satisfies HypothesisC there exists only one weak solution > to equation and we have that \|\\>=|\|>> and =S>\> (where is the only strong solution to equation and > is as in Theorem). <\proof> If satisfies HypothesisC, by Corollary, is invertible and by Theorem we have that > is unique and \|\\>=|\|>.> By Remark we have that >\>, where is the unique strong solution of, is the unique weak solution to the same equation. In this section we want to prove the dimensional reduction of equation, namely that the law in 0 of at least a (weak) solution to equation, has an explicit expression in terms of the potential . \; The original idea of Parisi and Sourlas for proving this relations was to transform expectations involving the solution > to equation (taken at the origin) into an integral of the form\ <\equation> \|)>|]>=hw|)>detUw|)>|)>eUw|)>,\w\-|Uw|)>|\<\|\|\>>>>\\, where is defined in equation . Then one can express the weight on the right hand side of as the exponential V|)>\x \\\|\>>> involving the superfield <\equation*> \,|\>|)>=\+\\+|\>|\>+\\|\>, (see Section and Section for a more precise description) constructed from the real Gaussian free field > over >, two additional fermionic (i.e. anticommuting) fields ,|\>> and the complex Gaussian field >. Introducing these new anticommuting fields it can be argued that the integral admits an invariance property with respect to supersymmetric transformations. This implies the dimensional reduction, i.e.\ <\equation> =h|)>eV|)>\x \\\|\>>\\=>h\\. Unfortunately this reasoning is only heuristic since the integral on the right hand side of is not well defined without a spatial cut-off, given that both the determinant and the exponential are infinite.\ For polynomial potentials , a rigorous version of this reasoning was proposed by Kleinetal.. More precisely Kleinetal. give a rigorous proof of the relationship introducing a suitable modification due to the presence of the spatial cut-off , but they do not discuss the relationship between equation and the reduction. \; In this section we do not want to propose a rigorous version of the previous reasoning which will be given in Section . Here we only assume that the conclusion of Parisi and Sourlas' formal proof holds for a general enough class of potentials. More precisely we assume Theorem below. For technical reasons, which will become clear in the following (see Remark below), in order to state Theorem , we need first to introduce an additional class of potentials. <\description-paragraphs> >>.>We have the decomposition <\equation*> V=V+\V,V=|)>,y=,\,y|)>\\, with \0> and > a bounded function with all bounded derivatives on >.\ In Section below we will exploit a supersymmetric argument, described briefly at the beginning of this section, for the family of potentials satisfying the more restrictive Hypothesis>> to prove that in this case a cut-off version of equation. <\theorem> Under the HypothesesCO and>> if is any real measurable bounded function defined on > then we have\ <\equation*> >hw|)>\\w|)>\\=Z>h\\, where =>>\\w|)>\\\0>. <\proof> The proof is given in Section below. In the rest of this section we want to show how to derive from Theorem the dimensional reduction result for the solution to the elliptic SPDE. More precisely the goal of the rest of this section is to prove the following theorem. <\theorem> Under the HypothesesCO andQC there exists (at least) one weak solution > to equation such that for any measurable bounded function defined on > we have\ <\equation> >hw|)>\w|)>\\>||>hw|)>\w|)>\\\>>|||>h\\>>>>> where =>\w|)>\\\0>. This result is very important since it implies Theorem and Theorem.\ <\render-proof|Proof of Theorem and Theorem> The relation can be expressed in the following more probabilistic way. Suppose that on a given probability space >,\>|)>>, the map :\\\>\\> gives the weak solution > of Theorem, namely that the law of the >-random variable -\|)>\,\|)>> is the measure >. Then we have that, for any real measurable bounded function defined on >,\ <\equation*> \>>|)>|)>|Z>|]>=>h\\, namely we have proven Theorem. If we assume HypothesisC then by Corollary, Corollary and Theorem there exists a unique strong solution satisfying and we have proven as a consequence Theorem. The proof of Theorem will be given in several steps of wider degree of generality with respect to the hypothesis on the potential . Before we prove an auxiliary result. <\lemma> Under the Hypothesis>> we have that\ <\equation> >g\T\\\=>g\\. where is any bounded measurable function defined on >. <\proof> Using the methods of Section we can prove that the map satisfies Hypotheses DEG1, DEG2, DEG3 of Appendix. The claim then follows from Theorem and Theorem below, where we can choose the function to be any bounded continuous function since \L|)>> under Hypothesis>>. <\proposition> Under the HypothesesCO and>> there exists at least one weak solution > to equation satisfying. <\render-proof> Proof <|render-proof> Let \L|\|>\\|)>> be the span of the two linear spaces ,\L|\|>\\|)>> where > is composed by the functions of the form T>, where is a measurable function defined on > such that T\L|\|>\\|)>>, and > is formed by the functions of the form w|)>\w|)>>, where is a measurable function defined on > such that w|)>\w|)>\L|\|>\\|)>>. Note that > and >, and so =span,|}>>, are non-void since, under the Hypotheses >> and CO (see Lemma below), \L|)>> and so T,hw|)>\w|)>\L|)>> whenever are bounded. Define a positive functional :\\> by extending via linearity the relations <\eqnarray> w|)>\w|)>|)>>|>|hw|)>\w|)>\\\>>|T|)>>|>|g\\.>>>> to the whole >. We have to verify that > is well defined and positive on .> Suppose that there exist functions and such that T=hw|)>\w|)>> then, by Lemma, we have <\equation> >g \\=>g\T*\*\\=>hw|)>\w|)>\\\. This implies that > is well defined on \> and so on >. Obviously > is positive on >, and, by Theorem we have <\equation> w|)>\w|)>|)>=>hw|)>\w|)>\\\=>>|>h\\\0>>>>> whenever , and so w|)>\w|)>>, is positive. This means that > is positive.\ For any T\>, by Theorem and Theorem, we have <\multline> |\|>=>g \\|\|>\>>|>|\|>N\\=>T\|\|>\\=|f\|\<\|\|\>>.>>>> On the other hand, if >, by relation, \|f\|\<\|\|\>>>. These two inequalities and the positivity of > imply, by Theorem8.31 of on the extension of positive functionals on Riesz spaces, that there exists at least one positive continuous linear functional on |\|>\\|)>>, such that => for any >. The functional defines the weak solution to equation we are looking for. Indeed, since is a continuous positive functional on |\|>\\|)>> there exists a measurable positive function L>|\|>\\|)>\L>\|)>> such that =>fB|\|>\\>. Since \L> by Lemma below, we have > and so =>1\\=1>. This implies, since the function is positive, that the >-finite measure \=B|\|>\\> is a probability measure. Furthermore, since > contains all the functions T>, where is measurable and bounded, equality implies that >|)>=\>. This means that > is a weak solution to equation. Finally since > contains all the functions of the form w|)>\w|)>> where is measurable and bounded on > the measure > satisfies the thesis of the theorem. Unfortunately we cannot repeat this reasoning for general potentials satisfying the weaker HypothesisQC since both Theorem and Proposition exploit an > bound on > (see Lemma below) that cannot be obtained for more general potentials. Thus the idea is to generalize equation without passing from equation. Indeed it is possible to approximate any potential satisfying HypothesisQC by a sequence of potentials |)>> satisfying Hypothesis>> in such a way that the sequence of weak solutions |)>> associated with |)>> converges (weakly) to a weak solution associated with the potential (see Lemma, Lemma and Lemma below). Since equation involves only integrals with respect to a weak solution to equation, we are able to prove that equation holds for any potential approximating its weak solution > by the sequence |)>> satisfying equation. \ Let us now set up the approximation argument, starting with a series of lemmas about convergence of weak solutions. <\lemma> Let |}>\>> be a sequence of continuous maps on > such that for any compact \> we have that \>T> is pre-compact and there exists a continuous map such that \T> uniformly on the compact subsets of \ >. Let > be a set of probability measures on > defined as follows <\equation*> \\> probability measure on > such that >|)>=\> for some i>>|}>. Then \\>|\>>, where the closure is taken with respect to the weak topology on the set of probability measures on >, is non-void and <\equation*> \\> probability measure on > such that >|)>=\>>|}>. <\proof> First of all we prove that > is pre-compact for any \>. This is equivalent to proving that the measures in > are tight. Let > be a compact set such that |)>\1-\> for a fixed \\1>, then \>T|)>|\>> is a compact set in >. Consider \\> then there exists > such that >\=\>. This implies <\equation*> \\\T|)>|)>\\|)>|)>\\|)>\1-\, for any \>. Since > are pre-compact, |\>> are compact and |\>\|\>> if j>. This implies that > is non-void. If we consider a \\> there exists a sequence > weakly converging to >, for +\>, such that ,\>|)>=\> and \+\>. Proving that >|)>=\> is equivalent to prove that for any > bounded function with bounded derivatives defined on > taking values in > we have g\T \\=g \\>. Let the compact set defined before, then there exists a \> such that K>|T>-T|\<\|\|\>>\\> and that >g\T \\->g\T\\|\|>\\>, for the arbitrary \\1>. This implies that <\eqnarray> >g\T \\->g \\|\|>>|>|>g\T \\->g\T \\|\|>+>>|||T-g\T>|)>\\|\|>+>>||||g|\<\|\|\>>>\+>g\T>\\->g \\|\|>>>||>|+|\g|\<\|\|\>>>\+|g|\<\|\|\>>>\.>>>> \ Since > is arbitrary, from this it follows that >g\T \\=>g \\>.\ <\remark> The proof of Lemma proves also that given any sequence of \\> there exists a subsequence converging weakly to \\>. <\remark> In the following we consider a sequence of functions > satisfying HypothesisQC. To each function > of the sequence it is possible to associate a map :\> defined by \f\Vw|)>> and the corresponding map :\> defined by =w+U>. <\lemma> Let |}>\>> be a sequence of potentials satisfying the HypothesisQC and converging to the potential , and such that V> converges uniformly to V> on compact subsets of >; moreover we assume that >, , V> and V> are uniformly exponentially bounded and there exists a common function entering HypothesisQC for |}>\>> and . Let >, be the maps on > associated with > and respectively as in Remark . Then the sequence |}>\>> satisfies the hypothesis of Lemma. \ <\proof> Note that the a priori estimates and in Lemma are uniform in \> since they depend only on the function and the exponential growth of ,V,\V,\V>. From this we can deduce the pre-compactness of the set \>T|)>> for any compact set \\> using a reasoning similar to the one proposed in Theorem and Theorem.\ Proving that > converges to uniformly on the compact sets is equivalent to prove that the map =f\Vw|)>> converges to =f\Vw|)>> in > uniformly on the compact subsets of >. Let be a compact set of >, then there exists an 0> such that w|\|>\M>|)>> (where we suppose without loss of generality that \1>). By hypotheses we have that there exist two constants ,\\0> such that V|\|>,V|\|>\e+\>>, thus there exists a compact subset > of > such that >|)>expM>|)>+2\|)>\x\\>, for some \>. Denote by >> the ball of radius > M>|)>> then we have\ <\eqnarray> K>|U-U|\<\|\|\>>>>|>|>|)>eM>|)>+2\>\x|\|>>>|||K>>|)>Vw|)>-\Vw|)>|\|>\x|\|>>>||>|+B>>V-\V|\|>|)>>|)>\x>>||>|,>>>> as +\.> This means that +\>K>|U-U|\<\|\|\>>>|)>\2\>, and since > is arbitrary in > the theorem is proved.\ <\lemma> Let be a potential satisfying HypothesisQC, then there exists a sequence |}>\>> of bounded smooth potentials converging to and satisfying the hypothesis of Lemma.\ <\proof> Let be a potential satisfying the HypothesisQC and let > the function whose existence is guaranteed by HypothesisQC. Let, for any \>, \supB>|\|>> and let \G>\V> where\ <\equation*> G\>|\k>>,>>||\k>>.>>>>> Let > be a smooth compactly supported mollifier and denote by >> the function >\\\>|)>.> We want to prove that =\\>>, for a suitable sequence \\>, is the approximation requested by the lemma. Without loss of generality we can suppose that > is a positive function depending only on the radius > and increasing as \+\>. Under these conditions, HypothesisQC is equivalent to say that for any unit vector \\> we have that for any \> <\equation*> max\\V|)>,0|)>\. We want to prove that |)>=+sup|)>|)>> is the function requested by the lemma. Since for any unit vector \\> we have \\|\|>\\\V|\|>> and since > is absolutely continuous we obtain <\equation*> -\\V|)>=\\|)>\\>|)>|)> <\equation*> \\\V+r|)>,0|)>\\>|)>\\\>. Furthermore we have that =V> on > and so there exists a sequence |}>> such that \0> and B>V-\V|\|>\>. Since > is smooth and bounded and <\equation*> \\>\+sup|)>|)>=H, we conclude the claim. Finally we are able to prove for all QC potentials, which will conclude this section. <\render-proof> Proof of Theorem <|render-proof> By Proposition the equality holds when satisfies the Hypothesis>> for some \0>, i.e. if =V,V>=V+\|)>> for some bounded potential >. It is clear that if \0> the potentials ,V>> converge to the potential > and the hypothesis of Lemma hold. This means that if |^>> is a sequence of probability measures such that |^>> is a weak solution to the equation associated with ,V>> satisfying the thesis of Proposition, by Remark and Lemma, there exists a probability measure |^>>, that is a weak solution to the equation associated with >, such that |^>\|^>> in the weak sense, as \> and \0>. We want to prove that |^>> is a weak solution to the equation associated with > satisfying equation . The previous claim is equivalent to proving that <\equation> <\multline*> >gw|)>efV,B>w|)>\x>\|^>\>>|>gw|)>efVw|)>\x >\|^>,>>>> as \0>, for any continuous bounded function , and that >\\> weakly, where \>=expV,B>|)>\x/Z>> and \=\\>=expV|)>\x/Z>. Proving relation is equivalent to prove that <\equation*> fV,B>w|)>\x\fVw|)>\x uniformly on compact sets of >. This assertion can be easily proved using the methods of Lemma. Indeed for any in the compact set >, using the same notations of the proof of Lemma, we have\ <\multline> fV,B>w|)>\x-fVw|)>\x|\|>\>>|\>|\|>>|)>|)>\x=C\\0.>>>> The weak convergence of >> to > easily follows from Lebesgue's dominated convergence theorem. \ The previous reasoning proves the theorem for any bounded potential >. Using Lemma we can approximate any potential satisfying HypothesisQC by a sequence of bounded potentials >. Using Lemma, Remark, Lemma and a reasoning similar to the one exploited in the first part of the proof we obtain the thesis of the theorem for a general potential satisfying HypothesisQC.\ Define <\equation> \\>hw|)>\w|)>|Z>\ \, with the notations as in Section (Theorem) and Section (Theorem). In this section we prove Theorem, i.e. the identity <\equation> \=>h\\. It is important to note that > appears without the modulus in.\ \; Let us start by unfolding the definition of > and w|)>> in to get the expression <\multline> *\=hw|)>det>+\U|)>\>>|exp-\<\|\|\>U\<\|\|\>>+4Vw|)>f\x|)>\\.>>>> In order to manipulate the regularized Fredholm determinant we approximate the right hand side by\ <\multline> >*\>\h>w|)>det>+\U>|)>\>>|exp>|)>-\<\|\|\>U>\<\|\|\>>+4V>w|)>f\x|)>\\.>>>> where \0> is a regularization parameter, >\\>=-\|)>>>, >> is the normalization constant such that >=1> and <\equation> U>\>\>\V>w|)>. We will prove below that \0>\>=\>. When \0>, U>=>\>\V>w|)>\>> is almost surely a trace class operator and >\>>. This means that we can rewrite the regularized Fredholm determinant > in term of the unregularized one (denoted by >) (see equation and the discussion in Appendix) obtaining <\equation> >*\>>||h>w|)>det>+\U>|)>\>>|||exp|U>,w|\>-\<\|\|\>U>\<\|\|\>>+4V>w|)>f\x|)>\\.>>>>> The determinant is invariant with respect to conjugation and so we can multiply U>> by +m|)>>> at the left hand side and by +m|)>>> at the right hand side (this last multiplication can be done since >> is a bounded operator from the Sobolev space ,2>|)>> into |)>> and +m|)>>> is a bounded operator from |)>> into |)>\W,2>|)>>). In this way we obtain <\multline> >+\U>|)>=det>+\\>f\V>w|)>\>|)>=>>|>+\f\V>w|)>\>|)>,>>>> where =>>, and featuring the operator f\V>w|)>\>>. Let > be the Gaussian measure given by the law of =\>w\|~>> under >. In other words the Gaussian measure > is the one whose Fourier transform is\ <\equation*> |~>>expk\\x|)>\\|)>=exp|>|\<\|\|\>>>|)>. \ The expression is then equivalent to <\equation*> h|)>det>+\f\V>w|)>\>|)>exp|\f\V|)>,-\|)>\|\>|)>\ <\equation*> \exp|2>\<\|\|\>\>f\V|)>\<\|\|\>>+4V|)>f\x|)>\\|)>. At this point we introduce an auxiliary Gaussian field > distributed as the Gaussian white noise > to write <\equation*> exp|2>\<\|\|\>\>f\V|)>\<\|\|\>>|)>=exp|f\V|)>,\>\|\>|)>\\|)>. We also introduce two fermionic fields ,|\>> realized as bounded operators on a suitable Hilbert space ,|\>>> with a state \|)>=|\|\>\|\>>> for which <\equation*> ,\|)>|}>=|\>,|\>|)>|}>=,|\>|)>|}>=0 <\equation*> ||\>|\>|)>|\>\|\>>=|\\|)>|\>\|\>>=0,|\|\>|)>|\>\|\>>=\\>|)>, where ,\|}>> is the anticommutator between bounded operators, i.e. ,K|}>=KK+KK> for any bounded operaotrs defined on ,|\>>>, and >> is the kernel of the operator >> (see Appendix for the definition of fermionic fields and Theorem for the existence of such fields). By Theorem and Remark, these additional fields allow to represent the determinant as <\multline> >+\f\V>w|)>\>|)>=>>||exp\f\\>V|)>|\>\x|)>|\>,|\>>.>>>> By tensorizing the fermionic Hilbert space ,|\>>> with the > space of the product measure \\> one can realize the fermionic fields ,|\>> and the Gaussian fields ,\> as operators on the same Hilbert space > with a state which we denote by |\|\>>> when this does not cause any ambiguity. As a consequence, we have <\equation> Z>\>=|h|)>exp>|)>|\>>, with an operator >> given by <\eqnarray*> >>|>|\f\V|)>|\>\x+>>||||f\V|)>,-\|)>\+i\>\|\>+4V|)>f\x.>>>> The operator satisfies the following important theorem. <\theorem> For all polynomials \\> and all 0> and all \0> we have <\equation> |p|)>>|)>|\>>=|p|)>P|)>|)>|\>>. This theorem is the key to our results and will be proved with the aid of supersymmetry in Section . Going back to equation a possible strategy would be to expand the exponential getting <\equation> |h|)>exp>|)>|\>>>||0>|h|)>>|)>|\>>>>>>> and then to use Theorem to prove that each average in the right hand side is equal to <\equation*> |h|)>V|)>|)>|\>>. Since <\equation*> |h|)>V|)>|)>|\>>=Z>>h\\, the equality would be proved taking the limit \0>. Unfortunately equation is not easy to prove since the series on the right hand side of does not converge absolutely for a general . For this reason we present below an indirect proof of. Given Theorem we will deduce \ Theorem from it via a sequence of successive generalizations.\ <\enumerate-alpha> First we consider potentials bounded and such that |\V|\<\|\|\>>>\m/2>; then the class of satisfying Hypothesis>> and ; finally those satisfying only>>. <\proposition> For all \\> bounded such that |\V|\<\|\|\>>>\m/2> and \\> bounded and measurable we have <\equation> |h|)>exp>|)>|\>>=|h|)>expV|)>|)>|\>> for all \0>. Let us introduce <\equation*> G>\|h|)>exp>|)>|\>>, <\equation*> H>\|h|)>expV|)>|)>|\>> for >. <\render-proof|Proof of Theorem> It is clear that >> is real analytic in >. By Lemma below the function >> is real analytic in >. It is enough then to prove G>=\H>> for any \>. Now <\equation*> \G>=|h|)>>|)>|\>>, <\equation*> \H>=\h|)>V|)>|)>\>. By the density polynomials in the space of two-times differentiable functions with respect to the Malliavin derivative (see Corollary1.5.1) we can approximate both G>> and H>> with expressions of the form |p|)>>|)>|\>>> and |p|)>P|)>|)>|\>>> where are polynomials and therefore conclude from that G>=\H>> for all 0>. The following two lemmas prove the claimed analyticity of >>. <\lemma> If is a bounded potential satisfying the Hypothesis, then >|)>|)>\L|)>> for any \1> and \.> Furthermore the integral exp>|)>|)>\\> is uniformly bounded for \> and in the compact subsets of >. <\proof> \ Under the Hypothesisof the lemma we have that\ <\equation*> |\U>|\<\|\|\>>\|\V|\<\|\|\>>>|m|)>>>, where |\|\<\|\|\>>> is the usual operator norm on >. PropositionB.8.1 of states that\ <\equation*> \\|)>|]>\|K|\<\|\|\>>>|)>|]>|)>>\|\K|\<\|\|\>>||\K|\<\|\|\>>|)>>|)>|]>|)>> whenever is a > map such that |\K|\<\|\|\>>\1>. Taking >> in the previous inequality we obtain the thesis. <\lemma> The function > is real analytic in >. <\proof> First of all we have that for any \> the map det\U>|)>=:D> is holomorphic in (see Theorem9.3). By Cauchy theorem this means that\ <\equation*> U>|)>|)>|\|>\\\>>|)>|\|>|R>. On the other hand we have for any \> <\equation*> |\|>\exp|\U>|\<\|\|\>>|)>\exp+|)>|\V|\<\|\|\>>>|)>, where \> is some positive constant depending on but not on . Thus we obtain <\equation*> U>|)>|)>|\|>\+|)>|\V|\<\|\|\>>>|)>|R>. With a similar reasoning we obtain a uniform bound of this kind for exp>|\|>|)>.> Finally we note that <\equation*> \U>|)>|)>|]>=r|n!>\>|)>|)>>|)>|)>|]>. By Lemma, we note that\ <\eqnarray> eU>|)>>|]>|\|>>||U>|)>>>|)>|)>|]>|\|>>>||>|>\|)>U>|)>>e|)>U>|)>>|]>\+\>>>> for any \1> and \\|2|\V|\<\|\|\>>>>-.> Using the previous inequality it follows that >> is real analytic in the required interval.\ <\proposition> We have that =H> for .> <\proof> By Proposition , we need only to prove that >\G> as \0>. Since >, >, |\|>>> are continuous with respect to the natural norm of > and the Hilbert-Schmidt norm on \\> (see Theorem9.2 for the continuity of > and Proposition1.5.4 for the continuity of >), and since >|)>|)>> is bounded uniformly in > (for small enough) we only \ have to prove that, for \0>, >\U> in > and U>\\U> in \\> for almost every \>. We present only the proof of the second convergence, the proof of the first one being simpler and similar. We have that <\equation*> \U>=V>w|)>\>h|)>, thus proving the convergence of U>> in \\> is equivalent to proving the convergence of -\|)>>> to -\|)>> in \\> and the convergence of V>w|)>> to Vw|)>> in |)>>. The first convergence follows from direct computation using the Fourier transform of this operators. The second convergence follows from the fact that is smooth with bounded derivatives, decays exponentially at infinity and >w> converges to w> pointwise and uniformly on compact sets since -\|)>>\id>>, weakly as bounded operator on |)>> and -\|)>> is a compact operator from |)>> into |)>>. >> and > \; Let > denote a bounded smooth potential with all its derivatives bounded. Introduce the following equation for =|\>+\\>: <\equation> -\|)>|\>+t f \V|\>+\\|)>=0. Denote by > the infimum on \> over the eigenvalues of the dependent matrix V|)>>, and with > the supremum on \> over the eigenvalues of the same matrix.\ For |\0|\|>>,|\0|\|>>|)> >we have that equation has an unique solution that, by the Implicit Function Theorem, is infinitely differentiable with respect to when \C>|)>>. Define the formal series <\equation> S\1>\> |\>|\|>|n!>r. \; <\lemma> Suppose that > is a bounded function with all derivatives bounded such that\ <\equation*> |\V|\<\|\|\>>>\Cn!, where the norm is the one induced by the identification of V> as a multilinear operator and for some \>, then the power series > is holomorphic for any |\0|\|>>,|\0|\|>>|)>.> Furthermore the radius of convergence of > can be chosen uniformly for in compact subsets of |\0|\|>>,|\0|\|>>|)>.> <\proof> We define the following functions\ <\equation*> >\0>|\V|\<\|\|\>>>|n!>r,>\0>|\V|\<\|\|\>>>|n!>r. We have that the partial derivative |\>> solves the following equation <\eqnarray> -\|)>\|\>+t \V|\>|)>\\|\>>||V|\>|)>+t \V|\>|)>\\|\>|)>+>>|||V|\>|)>\\|\>>>>> Using a reasoning similar to the one of Lemma, it is easy to prove that\ <\equation*> |\|\>|\<\|\|\>>>\|-\V|\>|)>+t \V|\>|)>\\|\>|)>+t \V|\>|)>\\|\>|\<\|\|\>>>|m->\0|)>>, where it is important to note that the right hand side of the previous inequality depends only on the derivatives of order at most . The previous inequality and the method of majorants (see ) of holomorphic functions permit to get the following differential inequality for > <\equation> ->\0|)>-r >|)>|)>\|)>\>|)>. From the previous inequality we obtain that > is majorized by the holomorphic function > that is the solution of the differential equation (where the symbol > is replaced by ) depending parametrically on with initial condition =0>. Since > is majorized by > or by > if \k> the thesis follows.\ <\remark> An example of potential satisfying the hypotheses of Lemma is given by the family of trigonometric polynomials in >. <\lemma> Under the hypotheses of Lemma with > and assuming that is an entire function we have that =H> for any |\0|\|>>,|\0|\|>>|)>>. In other words the thesis of Theorem holds if =0>, > satisfies HypothesisC as well as the hypotheses of Lemma. <\proof> By Proposition we need only to prove that > is real analytic in the required set. By Corollary we have that\ <\equation*> G=\\+|\>|)>et V\+|\>|)>f\x>|]>. Then the thesis follows from Lemma and the analyticity of and of the exponential. Let be a potential satisfying the Hypothesis>> then there exist > such that +\V> and we define <\equation*> V>=t V+\V, for any \>. Denote by >> the corresponding map from > into >. Let \\> be a continuous bounded function. We write\ <\equation*> G>\>hw|)>det>+\U>|)>\ <\equation*> \exp>|)>-\<\|\|\>U>\<\|\|\>>+4>V>w|)>f\x|)>\\ and <\equation*> H>\Z>hexp|2>+t V+\V|)>|)>\y. It is evident that the thesis of Theorem is equivalent to prove that <\equation*> G>=H> for any bounded potential >, any continuous and bounded and any |\0|\|>>,|\0|\|>>|)>>. This fact is the result of the next proposition. <\proposition> Under Hypothesis>> we have that >=H>> for any |\0|\|>>,|\0|\|>>|)>>. In other words the thesis of Theorem holds if satisfies also HypothesisC. <\proof> By Lemma we know that Theorem holds for any =0> and for any bounded potential satisfying HypothesisC and the hypothesis of Lemma. Thus if we are able to approximate any potential satisfying Hypothesis>> and HypothesisC by potentials of the form requested by Lemma the thesis is proved. We can use the methods of the proof of Lemma for approximating a potential satisfying Hypothesis>> by a sequence of potentials > satisfying the hypothesis of Lemma. More in detail, using the notations of Lemma, we have that the sequence of functions > is composed by smooth, bounded functions and, if satisfies Hypothesis>>, they are identically equal to outside a growing sequence of squares \\>. This means that >, which is the periodic extension of > outside the square >, is a smooth function for any \>. Since > is periodic it can be approximated with any precision we want by a trigonometric polynomial >. Furthermore since satisfies HypothesisC, also > satisfies HypothesisC and we can choose the trigonometric polynomial > satisfying HypothesisC too. In this way we construct a sequence of potentials =P> satisfying the hypotheses of Lemma and converging to uniformly on compact sets. Thus the thesis follows from Lemma, Lemma, Corollary and the fact that the functions of the form \+|\>|)>>, where is an entire function, are dense in the set of measurable functions in \+|\>> with respect to the |)>> norm. >>> <\lemma> Under the Hypothesis>> we have U|)>\L>|)>>. <\proof> We follow the same reasoning proposed in for polynomials. First of all, by the invariance property of the determinant with respect to conjugation, we have that <\equation*> detU|)>=det|)> where > is the selfadjoint operator given by <\equation> O=-\|)>>\>Vw|)>\-\|)>>h|)>. Since satisfies the HypothesisQC the eigenvalues of the symmetric matrix V> (where \>) are bounded from below. Furthermore we can write the matrix V> as the difference of two commuting matrices V=V-V> where ,V> are symmetric, they have only eigenvalues greater or equal to zero and \ker V=>. We denote by ,O> the two operators defined as in equation replacing V> by > and > respectively. Obviously > and > are positive definite and -O>. By Lemma3.3 we have that\ <\equation*> |)>|\|>\exp|O|\<\|\|\>>|)>. Using a reasoning similar to the one of Proposition and the fact that, under the Hypothesis>>, the minimum eigenvalue > of V> has a finite infimum >> that is the same as the one for > we obtain <\equation*> U|)>|\|>=|)>|\|>\exp|f|\<\|\|\>>|)> for some positive constant . In particular we have U|)>\L>.> In order to prove that |)>\L> we split into two pieces. First of all if > is the minimum eigenvalue of V> we recall that =inf\>\>. Moreover we shall set <\equation*> >\U-\0|)>f*\, and \U->>. We also set |2>>. We introduce a useful approximation of >> for proving Theorem. Let > the projection of an |)>> function on the momenta less then , i.e. <\equation*> P=\n>ex>\k, where > is the Fourier transform of defined on >. We can uniquely extend the operator > to all tempered distributions. In this way we define > as <\equation> U\PVPw|)>|]> We shall denote by >> the expression corresponding to where is replaced by . <\lemma> Under the Hypothesis>> there exist two positive constants > independent on 2> and \> such that\ <\equation> \>->|)>|\|>|]>\Cn>. Furthermore a similar bound holds also for \U\<\|\|\>-\<\|\|\>\U\<\|\|\>|\|>|]>> and |\w\<\|\|\>>-\<\|\|\>Pw|)>|\<\|\|\>>>|\|>|]>>. <\proof> First of all we write >=U+>> where =f \Vw|)>>, and we consider the corresponding decomposition for >>. If we prove that an inequality analogous to holds for -U> and >->> separately then the inequality holds. In order to prove the lemma we use the following inequality (proven in PropositionB.8.1) <\equation> <\aligned> >|2>\|)>|]>\>||K|\<\|\|\>>>|)>|]>|)>>\>>|||1-\c>|\K|\<\|\|\>>|)>|]>|)>>>>>> that holds when |\K|\<\|\|\>>\L>>, |\K|\<\|\|\>>\c\1> and \\>>. Putting |\>-U|)>> for |\>> small enough, since |\-U|)>|\<\|\|\>>,\<\|\|\>\-U|)>\<\|\|\>\L>> with a bound uniform in , we have that <\equation> <\aligned> \-U|)>|)>|]>\>||U-U|\<\|\|\>>>|)>|]>|)>>\>>|||\-U|)>|\<\|\|\>>|)>|]>|)>,>>>> for suitable ,\\0> and for all \>. First of all we want to give a bound for the right hand side of providing a precise convergence rate to the constant 1 of the upper bound for the right hand side as +\>. We first note that <\equation> \|U-U|\<\|\|\>>>|)>|]>=>|k!>\|U-U|\<\|\|\>>>|]>. Using a reasoning like the one in the proof of Proposition we have that\ <\equation*> |U-U|\<\|\|\>>>\|\V|\<\|\|\>>>|Q|\<\|\|\>>>+|\V|\<\|\|\>>>>Qw|)>|)>\x, where =I-P>. From the previous inequality and the hypercontractivity of Gaussian random fields we obtain that\ <\eqnarray*> |U-U|\<\|\|\>>>|]>>|>||Q|\<\|\|\>>>+>f\w|)>|)>|]>\x|)>>>||>||Q|\<\|\|\>>>+k |f|\<\|\|\>>\w|)>|)>|]>,>>>> where the constants implied by the symbol > do not depend on . The right hand side converges then for +\> to 1 as we have announced. Using the Fourier transform, the fact that is a Schwartz function, and the fact that w> is equivalent to a white noise transformed by the operator -\|)>> it is simple to obtain that |Q|\<\|\|\>>,\w|)>|)>|]>\>.> Then using the fact that \Ck!> and inserting the previous inequality in equation we obtain <\equation*> \|U-U|\<\|\|\>>|)>|]>\1+C|n>|\|n>|)>>, that holds when \0> is small enough and for two positive constants ,C>. Using similar methods it is possible to prove a similar estimate for |\-U|)>|\<\|\|\>>|)>|]>>. Inserting these estimates in the inequality , we obtain <\equation> \\-U|)>|)>|]>-1\|n>, where the constants implied by the symbol > do not depend on and on >, when > is smaller than a suitable \0>. Using the inequality we obtain that <\multline> >\|!>\-U|)>|)>|]>=>>|>n>\-U|)>|)>|]>-1|)>\>|n>>\+\.>>>> Since the terms of an absolutely convergent series are bounded we obtain <\equation*> \-U|)>|)>|]>\!|\n>>\n>. Using Young inequality we obtain that the inequality holds for any 2>. The estimate for >->|)>> follows from the fact that >> is a polynomial of at most third degree and from hypercontractivity estimates for polynomial expressions of Gaussian random fields. The result for |\U|\<\|\|\>>-|\U|\<\|\|\>>> can be proved using the same decomposition of and > and following a similar reasoning. The result for |f\w\<\|\|\>>-\<\|\|\>f Pw|)>|\<\|\|\>>>|\|>|]>> can be proved using hypercontractivity for polynomial expressions of Gaussian random fields. \ In the following we write =Tr\\|)>>. It is important to note that\ <\equation*> c=\n>+m>\x\log, where the integral is taken on the ball \n> on >. <\lemma> There exists a \0> depending only on and > such that for any \\\> and satisfying the Hypothesis>> there exist some constants ,C,C\0> such that\ <\equation*> \>|)>-R>f\w|)>\x-\<\|\|\>\U\<\|\|\>\-C-Cc> for any \>. <\proof> If K|\|>|)>\+\> and \> we have that =\K,w\>-TrK|)>>. Using this relation we obtain that\ <\multline*> >|)>=>P>W\w|)>|)>w \x+>>|Tr>\>W\w|)>\P-\|)>|)>|)>>>>> From this we obtain the lower bound <\eqnarray> >P>W\w|)>|)>w\x>||>f\>W\w|)>-\|)>\w|)> \x>>|||>f\>Ww|)>-\|)>w|)> \x>>|||>f\\>Ww|)>\\w\\\w*\x+>>|||>f\w\>Ww|)>\x+>>|||>f|)> Ww|)>\x>>||>|>f\w\>Ww|)>-bWw|)>|)>\x >>>> On the other hand we have\ <\equation*> Tr>\>Ww|)>\P-\|)>|)>|)>=c>\\>Ww|)>f \x <\equation*> \|p>+>\>Ww|)>w|)>|)>f \ x, where +=1> and 2>. Furthermore we have that\ <\multline> |\U|\<\|\|\>>\>+m|)>>\x>\>Vw|)>|)>f \x=>>|>\>Vw|)>|)>f \x,>>>> where =>+m|)>>\x>. Using the previous inequality we obtain that <\eqnarray> ||>|)>-R>fw|\|>\x-|\U|\<\|\|\>>>>|||-|p>+>f\w\>Ww|)>-bWw|)>|)>\x+>>|||->f\>Ww|)>|)>|q>+\\>w|)>|)>+Rw|\|>|)>\x >>>> It is simple to see that there exists a \0> (depending only on > and >) such that for any potential satisfying the Hypothesis>> with \\> the expression <\equation> my\>W-bW-\>W|)>|q>-\\>Vw|)>|)>-R is bounded from below and thus the thesis of the lemma holds. <\remark> Lemma Lemma are the only places where HypothesisCO and Hypothesis>> are used in an essential way.\ Indeed we are able to obtain the estimate , using the technique of the proof of Lemma , only if is a sum of a bounded function and a polynomial. Furthermore we can obtain that the expression is bounded from below, for > small enough and for any 0>, only if the expression \>W> is positive at infinity and it is able to compensate the growth of all the other terms in expression . The previous conditions are satisfied only if \4m> and is a sum of a bounded function and a polynomial of fourth degree (not less because of the presence of >, and no more since in the other cases the growth at infinity of \>Vw|)>|)>> would have been strictly stronger than the growth at infinity of \>W>). This is the main reason for the restriction on > in HypothesisCO and for the special form of required by Hypothesis>>. \ <\lemma> Given a |)> >there is a 0> big enough such that <\equation*> exp|)>-R>fw|\|>\x|)>\L|)>. <\proof> This lemma is proven in Lemma3.2.\ <\lemma> Suppose that satisfies the Hypotheses CO, then there exists \0> depending only on and > such that for any \\> and any satisfying the Hypothesis>> we have that <\equation*> exp+|\U|\<\|\|\>>|)>|)>\L|)> for any |)>.> <\proof> The thesis follows from Lemma, Lemma, Lemma and Lemma using a standard reasoning due to Nelson (see LemmaV.5 of or ) due to the fact that from the previous results it follows that there exist two constants ,\\0> independent on such that <\equation*> \\\|\|)>\\|)>|}>|)>\e>>. <\render-proof|Proof of Theorem> By Proposition in order to prove the theorem it remains only to prove that >> is real analytic for any \>. The proof of this fact easily follows from Lemma exploiting a reasoning similar to the one used in Lemma. At this point our main result is reduced to check the claim of Theorem, namely that for all polynomials \\> and all 0> and all \0> we have the equivalence <\equation> |p|)>>|)>|\>>=|p|)>P|)>|)>|\>>. Since the expressions in the expectations are polynomials in the fields ,\,\\|\>> which are \Pfree\Q, namely satisfy either the bosonic or fermionic version of Wick's theorem (see, e.g., Chapter3 Section8) the claim could be checked by explicit computations. However this is still not trivial and a better understanding of the structure of the required computations can be obtained introducing a supersymmetric formulation involving the > and the >. This new formulation exposes a symmetry of the problem which is not obvious from the expressions we obtained so far. \; For an introduction to the mathematical formalism of supersymmetry see e.g.. The details of the rigorous implementation of the ideas exposed here is the main goal of the paper of Klein et al. and of the modifications we implement here in order to overcome a gap in their proof. Formally the superspace > can be thought as the set of points ,|\>|)>> where \> and ,|\>> are two additional anticommuting coordinates. A more concrete construction is to understand > via the algebra of smooth functions on it. \ Let ,\,\|)>> be the (real) Grassmann algebra generated by the symbols ,\,\>, i.e. ,\,\|)>=,\\,\\\,\,\\\|)>> with the relations \=-\\>.\ A >> function \\,|\>|)>> is just a quadruplet >,f>,f|\>>,f|\>>|)>\>|)>|)>>, via the identification <\equation> F=f>+f>\+f|\>>|\>+f|\>>\|\>. The function can be considered as a function \\> by formally writing <\equation*> F,|\>|)>=F. In particular we identify >|)>> with >;\,|\>|)>|)>>. >|)>> is a non-commutative algebra on which we can introduce a linear functional defined by <\equation*> F\F,|\>|)>\x\\\|\>\->f|\>>\x, where |\>>> as in equation , induced by the standard Berezin integral on > satisfying <\equation*> \\\|\>=\\\\|\>=|\>\\\|\>=0,\|\>\\\|\>=-1. <\remark> A norm on >|)>> can be defined by <\equation*> |F|\<\|\|\>>|)>>=sup\>>|\|>+>|\|>+|\>>|\|>+|\>>|\|>|)>, and an involution by <\equation*> >,|\>|)>=>|\>+>|\>\+|\>>|\>|\>+|\>>|\>\|\>, where the bar on the right hand side denotes complex conjugation. Given C;\|)>> we define the composition F:\\\> by <\equation*> r,|\>|)>|)>\r>|)>+r>|)>f>\+r>|)>f|\>>|\>+r>|)>f|\>>\|\>, in accordance with the same expression one would get if were a monomial. Moreover we can define similarly the space of Schwartz superfunctions |)>> and the Schwartz superdistributions |)>=\;\,|\>|)>|)>> where \|)>> can be written >+T>\+T|\>>|\>+T|\>>\|\>> with >,T>,T|\>>,T|\>>\\|)>> and duality pairing <\equation*> T=-T>|\>>|)>+T>|\>>|)>-T|\>>>|)>-T|\>>>|)>,f>,f>,f|\>>,f|\>>\\|)>. We take the generators ,|\>> to anticommute with the the fermionic fields ,|\>>, and introduce the complex Gaussian field <\equation*> \\-\-\|)>\+i>\|)> and put all our fields together in a single object defining the <\equation*> \,|\>|)>\\+ |\>\+\|\>+\\|\>, where \>. We also define <\eqnarray*> ,|\>|)>|)>>|||)>+\V|)>|\>\ +\|\>|)>+>>|||V|)>\+\V|)>\|\>|]>\|\>>>>> and since <\equation*> +4\|\>|)>=|)>+4|)>\|\>, where :\\\> is the smooth function such that =|)>> and =|)>> (see Section ), we observe that <\equation*> -V,|\>|)>|)>+4\|\>|)>\x \\\|\>=f\V|)>\\x+ <\equation*> +\V|)>\|\>+4V|)>f|]>\x=Q>. By introducing the superspace distribution |\>\x|)>> we have also, by similar computations: <\equation*> p|)>=-p,|\>|)>|)>\|\>\x|)>\\\|\>. As a consequence we can rewrite |p|)>>|)>|\>>> as an average over the superfield >: <\equation> <\aligned> >\>||p|)>>|)>|\>>=>>|||p,|\>|)>|)>\|\>\x|)>\\\|\>|)>|\>>>|||P,|\>|)>|)>+4\|\>|)>\x \\\|\>|)>|\>>>>>> While all these rewriting are essentially algebraic, the supersymmetric formulation makes appear a symmetry of the expression for >> which was not clear from the original formulation. In some sense the reader can think of the superspace ,|\>|)>> and of the superfield ,|\>|)>> as a convenient bookkeeping procedure for a series of relations between the quantities one is manipulating. \; A crucial observation is that the superfield > is a free field with mean zero, namely all its correlation functions can be expressed in terms of the two-point function |\,|\>|)>\,|\>|)>|\>>> via Wick's theorem. A direct computation of this two point function gives: <\align> |\,|\>|)>\,|\>|)>|\>>=>||\\|)>|\>>-||\>\|)>|\>>\|\>-|\|\>|)>|\>>|\>\>>|||\\|)>|\>>\|\>+|\\|)>|\>>\|\>+>>|||\\|)>|\>>\|\>\|\>>>||>|)>+\\>|)>|\>-|\>\|)>+>>||-\|)>\>|)>|\>+\|\>|)>+>>||-\|)>\>|)>-\>|)>|)>\|\>\|\>.>>>> Upon observing that -\|)>\>=\>>, -\|)>\>=\>> and that \ |\>+|\>\+\|\>+\|\>=-\|)>|\>-|\>|)>> we conclude <\equation> |\,|\>|)>\,|\>|)>|\>=C>,\-\,|\>-|\>|)> where <\equation*> C>,|\>|)>\\>-\\>\|\>. <\remark> Note that when =0>, the superfield > corresponds to the formal functional integral <\equation*> e-\|)>\|]>\x\\\|\>>\\ where \=\\\|\>\\\\> and where =\+\>\|\>>> is the superlaplacian, where >,\|\>>> are the Grassmannian derivative such that >|)>=\|\>>|\>|)>=-1>, >|\>|)>=\|\>>|)>=0>, >|\>\|)>=-|\>> and |\>>|\>\|)>=\> (see, e.g, Chapter 20 or Section 16.8.4). Then <\equation*> -\|)>\|]>\x\\\|\>=|\>-\|)>\-\\+2\-\|)>\|]>\x \; <\equation*> =-\|)>|\>+-\|)>\|)>+\|]>\x and this indeed corresponds to the action functional appearing in the formal functional integral for ,|\>,\,\|)>>. This is in agreement with the fact that the two point function satisfies the equation <\equation*> -\|)>C>,|\>|)>=\\|)>\|\>|)>, where \|)>\|\>|)>> is the distribution such that <\equation*> F,|\>|)>\\|)>\|\>|)>\x\\\|\>=f>, namely, >> is the Green's function for -\|)>>. On >|)>> one can introduce the (graded) derivations\ <\equation*> Q\2\\+x\|\>>,>\2|\>\-x\>, where \>, > (and in the following also =div\|)>>) acts only on the space variables \>,which are such that <\equation*> Q+4\|\>|)>=>+4\|\>|)>=0, namely they annihilate the quadratic form +4\|\>>. Moreover if > F=0>, for as in equation , then we must have <\equation*> 0=Q F,|\>|)>=2\\f>+x f|\>>+2\f|\>>\|\>- x f|\>>\ <\equation*> 0=> F,|\>|)>=2|\>\f>+x f>-2\f>\|\>+ x f|\>>|\> and therefore\ <\equation*> \f>= f|\>>f>=f|\>>=0. If we also request that is invariant with respect to > rotations in space, then there exists an such that |)>=f>> from which we deduce that |)>=\f|)>=\f>= f|\>>> which implies <\equation*> f+4\|\>|)>=f|)>+4f|)>\|\>=f>+f|\>>\|\>=F,|\>|)>. Namely any function satisfying these two equations can be written in the form <\equation*> F,|\>|)>=f+4\|\>|)>. Observe that if we introduce the linear transformations <\equation*> \>|)>>|>>||\>>>>>>=>\\ +2b|\>\>>|-b|)>\>>||\>+>|)>\>>>>>\,|\>,\|)> for >\\> and where > is a new odd variable anticommuting with ,|\>> and itself, then we have <\equation*> ||\t>|\|>\>|)>F,|\>|)>=||\t>|\|>F>|)>,|\>|)>|)>=>+>\Q|)>F,|\>|)> so >|)>=exp>+>\Q|)>> and >|)>\>|)>=\b,>|)>>. In particular C>|)>> is supersymmetric if and only if is invariant with respect to rotations in space and for any >\\> we have >|)>=F>. By duality the operators >> and >|)>> act also on the space |)>> and we say that the distribution |)>> is supersymmetric if it is invariant with respect to rotations in space and =>=0>. For supersymmetric functions and distribution the following fundamental theorem holds. <\theorem> Let |)>> and |)>> such that > is a continuous function. If both and are supersymmetric, then we have the reduction formula <\equation> T,|\>|)>\F,|\>|)>\x \\\|\>=4\*T>F>. <\proof> The proof can be found in, Lemma4.5 (see also for a general proof on supermanifolds). Let us note that <\equation*> Q C>,|\>|)>=> C>,|\>|)>=0, indeed we can check that <\align> \>=>|>k||)>> ex>|+|)>>>>>|||)>>>k||)>> ex>\ +|)>>>>>|||)>>>k||)>> \ ex>|+|)>>>=-|)>>\>>>|||2>\>>>>> As a consequence expectation values of polynomials over the superfield > are invariant under the supersymmetry generated by any linear combinations of >>.\ <\remark> The previous discussion implies that <\equation> \>|)>C>,|\>|)>=C>,|\>|)>. As a consequence, the superfield \\>|)>\> is a Gaussian free field and has the same correlation function >> as > given by equation. However it is important to stress that this does not imply that > has the same \Plaw\Q as >, namely that |F|)>|\>=|F|)>|\>> for nice arbitrary functions. Indeed the correlation function given in equations involves only the product |\,|\>|)>\,|\>|)>|\>> of the complex superfield > and not also the product |\,|\>|)>|\>,|\>|)>|\>> of > with its complex conjugate |\>>. The law of > would have been invariant with respect super transformations if and if only |\,|\>|)>\,|\>|)>|\>> and |\,|\>|)>|\>,|\>|)>|\>> had been both supersymmetric. Unfortunately the function |\,|\>|)>|\>,|\>|)>|\>> is not invariant with respect to super transformations. As explained in Remark, the law of > is not supersymmetric. Nevertheless we can deduce important consequences from the supersymmetry of the correlation function >>. More precisely, since > is a free field Wick's theorem (see, e.g., Chapter3 Section8) hold and\ <\eqnarray> ||\,\,|\>|)>|\>>=>>|,j|)>|}> >C>>-x>,\>-\>,|\>>-|\>>|)>,>>>>>>|>>|||\,\,|\>|)>|\>>=0.>|>>>> By the supersymmetry of >,\-|\>,\-|\>|)>> and of its products, we obtain that <\equation*> |\>|)>|)>,\,|\>|)>|\>>=|\,\,|\>|)>|\>>. The previous equality implies that\ <\equation> <\aligned> |P|)>\\>|)>|)>\x \\\|\>|\>>=>||\>|)>|)>|)>\F\x \\\|\>|\>>>>|||P>|)>|)>|)>\F\x \\\|\>|\>>>>|||P|)>\F\x \\\|\>|\>>,>>>> where ,\,P> are arbitrary polynomials and ,\,F> arbitrary functions on superspace. <\lemma> Let ,\.,F\|)>> be supersymmetric smooth functions and ,\,P> be polynomials then <\equation*> |P|)>,|\>|)>\F,|\>|)>\x \\\|\>|\>>=|)>|f>P|)>|\>>. <\proof> We define the distribution \\|)>> in the following way: <\multline*> \|P|)>,|\>|)>\G,|\>|)>\x \\\|\>|\>\>>||\P|)>,|\>|)>\F,|\>|)>\x \\\|\>|\>>>>>> for any \|)>>. Using the fact that ,\,F> are supersymmetric and relation we have that <\eqnarray> |>|)>|)>=|P|)>\\>|)>\x \\\|\>P|)>\F\x \\\|\>|\>>>|>>|||P|)>\\>|)>\x \\\|\>P|)>\\>|)>|)>\x \\\|\>|\>>=.>|>>> This means that > is supersymmetric and since > is also supersymmetric, by Theorem we conclude <\multline*> |)>=f>=>>||)>|f>P|)>F\P|)>\x \\\|\>|\>>=>>>> where |)>\x|)>\|\>>. Setting <\multline*> \|P|)>K\x \\\|\>|)>|\>\>>||\P|)>G\x \\\|\>|)>P|)>F\x \\\|\>|\>>>>>> and reasoning similarly we also conclude that |)>=\>. Proceeding by transforming each subsequent factor, we can deduce that <\multline*> |P|)>F\x \\\|\>|\>>=>>||P|)>K\x \\\|\>|\>>=|)>|f>P|)>|\>>.>>>> <\render-proof|Proof of Theorem> It is enough to use Lemma with =p>, =\=P=P>, \ =-\|\>\> and =\=F=+4\|\>|)>> to conclude. <\remark> The dimensional reduction proof via supersymmetry is already present in and indeed our result is analogous, under different hypotheses, to TheoremII in. The proofs of Lemma, Lemma and Lemma above follows the same ideas of Lemma3.1, Lemma3.2 and Lemma3.3 in. We decided to propose a detailed proof of Theorem mainly for two reasons: <\enumerate-numeric> The first reason is that the hypotheses on the potential of Theorem and of TheoremII in are quite different. Indeed in only polynomial potentials are considered while Hypothesis>> permits to consider polynomial of at most fourth degree perturbed by any bounded function. In order to prove the boundedness of > in |)>> under these different hypotheses \ we need to prove Lemma which is a trivial consequence of hypercontractivity when the potential is polynomial but is based on the non-trivial inequality (proven in) for general potentials . The second main reason is the difference in the use of supersymmetry and of the supersymmetric representation of the integral. Indeed, in our opinion there is a little gap in the proof of TheoremIII of that cannot be fixed without developing a longer proof. More precisely, in the proof of TheoremIII of it is tacitly assumed that the expression <\equation*> \\|g|)>expV|)>F\\\|\>\x|)>|\>>, is supersymmetric with respect to the function , i.e. if is a smooth function in >|)>> and >|)>> is a supersymmetric transformation, then we have that >|)>|)>=G>. In our opinion this fact is non-trivial since the law of > is not supersymmetric (see Remark). What can be easily proven is only that the expressions <\equation*> \\|g|)>V|)>F\\\|\>\x|)>|\>> are supersymmetric in (see Theorem above). This fact alone does not easily imply that > is supersymmetric. Indeed for the discussion in Section, we cannot guarantee that the series, which is equivalent to =0>\>, converges absolutely when growth at infinity at least as a polynomial of fourth degree (and we do not know under which conditions on and it converges relatively). In order to overcome this problem we propose a proof of Theorem which exploits only indirectly the supersymmetric representation of the integral in a way which permits to use only the supersymmetry of the expressions > and avoiding the proof of the supersymmetry of the expression >. In this section we prove Theorem on the removal of the spatial cut-off in the setting of HypothesisC. It is important to note that, differently from Theorem, we explicitly require that the potential satisfies HypothesisC and not only HypothesisQC. This is not due to problems in proving the existence of solutions to equation or in proving the convergence of the cut-offed solution to the non-cut-offed one without the HypothesisC (see Lemma). The main difficulty is instead to prove the convergence of |)>/Z> to . Indeed the previous factor does not actually converge and what we can reliably expect is that\ <\equation> lim1>Z\|)>\|\|)>|]>\1, where hereafter > denotes the solution to the equation with cut-off , i.e. |)>/Z> becomes independent with respect to the >-algebra generated by >.\ To prove directly is quite difficult due to the non-linearity of the equation or equivalently to the presence of the regularized Fredholm determinant in the expressions and (which is a strongly non-local operator). For this reason we want to exploit a reasoning similar to the one used in Section. With this aim we introduce the equation\ <\equation> -\|)>\+t f \V|)>=\ and the functions <\equation*> F\Z\|)>e>fV|)>\x>|]>, where is taken such that V+m> is positive definite, and =\|)>|]>> (where > is the solution to with 1>). By Lemma (whose proof does not use in any point the cut-off ) > is real analytic whenever is a trigonometric polynomial, V+m> is definite positive for any \> and is an entire bounded function. Furthermore, by Theorem, =H> (where =L\\>, see Section) which is real analytic. Thus if we are able to prove that 1>\F=\F> we have that =F> whenever V+m> is definite positive proving that Theorem when is a trigonometric polynomial satisfying HypothesisC. The idea, then, is to apply a generalization of Lemma, Lemma, Lemma and the reasoning in the proof of Proposition and in the proof of Theorem in order to obtain Theorem. <\remark> HypothesisC is required in an essential way in the proof of the holomorphy of >, in particular in Lemma. The fact that the cutoff is removed does not allow to reason by approximation as we did in Theorem.\ Since the proof is composed by many steps which are a straightforward generalization of the results of the previous sections of the paper, we write here only some details of the parts of the proof of Theorem which largely differ from what has been obtained before. \ Hereafter we denote by >> the function <\equation*> \>\exp|)>>|)> and introduce the space >> where \0> in the following way <\equation*> \>\+1|)>C>;\|)>, where >> is the space of continuous function with respect to the weighted >> norm <\equation*> |g|\<\|\|\>>,exp\>\sup\>>g|\|>. The triple >,\,\|)>> is an abstract Wiener space. We introduce the obvious generalization of equation <\equation> -\|)>|\>+\V|\>+\\|)>=0, where |\>=\-\\.>\ Now we want to prove a result that can replace Lemma. Indeed Lemma plays a central role in the previous sections of the paper, allowing to prove the existence of strong solutions to equation , the characterization of weak solutions in Theorem and Theorem and finally allowing to show the convergence of weak solutions using the convergence of potentials in Lemma, Lemma. <\lemma> Suppose that satisfies the HypothesisQC and suppose that |\>> is a classical solution to equation , then there exists a > depending only on > such that, for any \\>\ <\equation> ||\>|\<\|\|\>>,exp\>\|exp\|\|>|)>|\<\|\|\>>,exp\>, where |exp\|\|>|)>|\<\|\|\>>,exp\>> is almost surely finite and the constants implied by the symbol > depend only on and >. Furthermore for any open and bounded we have\ <\equation> ||\>|\<\|\|\>>>>\|expp\|\|>|)>|\<\|\|\>>>,\>expp||\>|\<\|\|\>>,exp\>|\>|\<\|\|\>>>,\>|)> where >\\\|\y\U,\\|}>> and \0>. <\proof> The proof is very similar to the proof of Lemma. We report here only the passages having the main differences. For any \0> there is a >\0> and for any \\>> \ we have <\equation*> >|)>|\>>-\>|\|>|\>>|\|>\\,x\\. Without loss of generality (using the result of Lemma) we have that \>|\>|\|>\>=0> and so |\>|\|>\>> has a positive maximum at >\\.> This means that |\>|\|>\>|)>>|)>\0> and |\>=-|\>|2\>>\\>> we have that\ <\equation*> -\|)>|\>>|)>|\|>\>>|)>\|\>|\|>\>|)>>|)>|2>+m|\>>|)>|\|>\>>|)> <\equation*> \-\>>|)>|\>>|)>\\V\>|)>+|\>>|)>|)>|)>. Using a reasoning similar to the one of Lemma the thesis follows. Since the bounds and in >> and >> imply the compactness in >> when \\>, Lemma permits to prove the existence of strong solutions to equation , their uniqueness when satisfies HypothesisC and the generalization of Lemma, Lemma, Lemma, Proposition and Theorem needed in order to prove Theorem.\ \; At this point the proof of Theorem requires only the following additional statement. <\theorem> Let be a trigonometric polynomial, let be a polynomial and let > be a sequence of cut-offs satisfying HypothesisCO, such that \1> on the ball of radius \> and such that \Cexp-r|)>|)>> for some positive constants C\\> independent on , then <\equation*> \H=lim+\>\F>=\F. To make the proof easy to follow we restrict ourselves to the scalar case, i.e. the case where . The general case is a straightforward generalization. We will also need certain results about iterated Gaussian integrals. So let us introduce first some notations.\ \; We denote by > the set of rooted trees with at least a external vertex which is not the root. We denote by > the tree with only one vertex other than the root. In this set we introduce two operations: if \> we denote by |]>> the tree obtained from > by adding a new vertex at the root which becomes the new root, and if \>> we denote by \\> the tree obtained by identifying the root of > and >. It is possible to obtain any element of > by applying iteratively a finite number of times the previous operations to >. Furthermore we define >\C|)>> by induction in the following way <\equation*> >\\\,|]>>\>f>\ y, <\equation*> \\>\>\>, where > is the Green function of the operator =-\|)>>. We need also to introduce the following notation. Suppose that ,\\> and let ,\>> be the set of all possible pairing between the external vertices (excepted their roots) of the forest \\> and let ,\>\,\>> the set of all possible pairing involving separately the vertices of > and >. If \> we write\ <\equation*> ,f>,\>=\|~>,f>>\|~>,f>>|]>, where |~>,f>>,|~>,f>>> are the expression >> where > is replaced by some copies of Gaussian white noises > one for each \ vertex of > and > which have correlation if |)>>\> and are identically correlated otherwise. <\lemma> With the notations and the hypotheses of Theorem we have that for any ,\\> <\multline*> +\>>>\f>>\x|]>+|\>>>||-\>>|]>\\f>>\x|]>|)>=0.>>>> <\proof> We present the proof only for the case , since the general case is a straightforward generalization. Since >>> are Gaussian random variables depending polynomially with respect to the white noise >, using the notation previously introduced we have\ <\multline*> >>\f>>\x|]>-\>>|]>\\f>>\x|]>=>>|\>,\>\>,\>>>,f>,\>f\x.>>>> Let us consider the simplest case when =\:=|]>\|]>> times and =\>=|]>\|]>> > times. In this case we have\ <\multline*> ,f>,\>=|)>f|)>\-x|)>\>>|-x|)>f|)>\f>|)>>-x|)>\ y\\ y\x\\x.>>>> In particular, since =\>, which is the Green function of =-\|)>>, is bounded and positive, and since > is positive we obtain that <\equation*> ,f>,\>|\|>\\\\|\>times>=> \ x>|+m|)>>>\l. Thus we get\ <\equation*> ,f>,\>|\|>\+1|)>\>+1|)> \ x>|+m|)>>>\l.|\|>\C, where \\>. Thus\ <\multline*> >,f>,\>f\x\>>|>|+1|)>>Cexp-r|)>|)>\x\,C,C>+1>\0.>>>> For the general case let us note that ,f>,\>> is built by taking the product and the convolution with the functions ,f> and =\>. We note that > appears one time for every pair of vertices ,V|)>\\>. Then, since >,\>> there is at least a couple |)>\\> such that is a vertex of > and > is a vertex of >. Now we can bound the function > with a constant > for all pairs of vertices ,V|)>|)>> and > by 1 obtaining, for any \>, that\ <\equation*> ,f>,\>\C>>>,\> for some ,k,k\\>. The thesis follows from the previous inequality and the bounds obtained on >>,\>.> <\render-proof|Proof of Theorem> We write\ <\equation*> >\L,t>|)>>\exp>fV,t>|)>\x |)>. We have\ <\eqnarray*> F>>||l\k>*\>>|\>|\>|]>>|)>|\|>|]>>>|||>>|]>+l\k>p\l-1>>>\>>|]>+|\>>>||||-\>>|]>\>>|]>|)>\|\>|]>>|)>|\|>,>>>> where we used the Leibniz rule for the derivative of the product and the relation\ <\equation*> |\>|]>>|)>|\|>=-p\l-1>\>>|]>\|\>|]>>|)>|\|>. Since |\>|]>>|)>|\|>> is bounded from above and below when +\> if we are able to prove that >>|]>\\F> and >>\>>|]>-\>>|]>\>>|]>\0> the theorem is proven.\ First of all we note that\ <\equation> -\|)>\\,t>=k f\,t>|)>|)>|\|> for 0> and ,0>=\\> for . This means that >>,>>> are given by a finite combination of convolutions and products between the function > (i.e. the Green function of >), the functions of the form >,0>|)>> (where >> is the -th derivative of ), the cut-off > and >. Since is a trigonometric polynomial, by developing and its derivative by Taylor series, we obtain the following formal expressions <\equation> <\aligned> >>=>|\>A>>>,>>|>>=>|,\,\\>B,\,\|)>>>f>>\x.>>>> The previous series are not only formal but they are actually absolutely convergent series. Furthermore we can change the integral, the expectation and the limit with the series. In order to prove this we now note that there exist two positive constants \0> such that the function is majorized (in the meaning of the majorants method) by x|)>> and let > be the polynomial which majorizes the polynomial . We now consider |~>>=,f>|)>> and |~>>=>fexp\,t>|)>\x|)>>. For what we said, |~>>>> and |~>>>> are a finite combination of convolutions and products between >, the functions of the form >,0>|)>> (where >> is the -th derivative of ), the cut-off > and >. Let |^>>> and |^>>> be some random variables having the same expression of |~>>>> and |~>>>> where we replace every appearance of ,0>|)>> by ,0>|\|>|)>>, every appearance of ,0>|)>> with exp,0>|\|>|)>> and so on. We introduce the following functions dependent on \> and defined recursively as follows <\equation*> >>\>>|\|>|]>>>\>f>>\ y <\equation*> \\>>\>>\>>. We, then, obtain that\ <\equation*> >>>=\>>>>>>>=,\,\\>,\,\|)>>>f>>\x. By our construction we have that >,,i>> are all greater or equal than zero and also the following inequalities hold >|\|>\>,,\,\|)>>|\|>\,\,\|)>>>. Furthermore we have >>|\|>\>>>. Finally >>>|\|>|]>,\>>>|\|>|]>> are finite for any , since the ,\,x> function\ <\equation*> \\,0>|)>|\|>|)>|]>\+\, for any \0>. Since > is positive the bounds on >>>|\|>|]>,\>>>|\|>|]>> can \ be chosen uniformly on . This implies that the series are absolutely convergent and by Lebesgue's dominated convergence theorem we can exchange the series with the summation and the limit. This means that\ <\eqnarray*> |+\>\>>\>>|]>-\>>|]>\>>|]>=>|>||+\>\>,\,\,\\>A>B,\,\|)>>>>\f>>\x|]>|\>+>|>|||-\>>|]>\\f>>\x|]>|)>=>|>||\>,\,\,\\>A>B,\,\|)>>lim+\>>>\f>>\x|]>+|\>>|>|||-\>>|]>\\f>>\x|]>|)>=0,>|>>> where in the last line we used Lemma. In a similar way it is simple to prove that <\equation*> \>>|]>\\F, and this concludes the proof. <\render-proof|Proof of Theorem> Using the bounds and we can prove the existence of strong solutions to equation , and their uniqueness when satisfies HypothesisC.\ Furthermore using again the bounds and and a suitable generalization of Lemma, Lemma, Lemma we can prove that Theorem holds for any potential satisfying HypothesisC if and only if Theorem holds for trigonometric potentials satisfying HypothesisC. \ \ The fact that Theorem holds for trigonometric potentials, satisfying HypothesisC, is a consequence of Theorem. This appendix summarizes the results of which are used in the paper and establish the related notations. Hereafter we consider an abstract Wiener space |)>> where is a separable Banach space, is an Hilbert space densely and continuously embedded in (with inclusion map denoted by W>) called Cameron-Martin space and > is the Gaussian measure on associated with the Cameron-Martin space, i.e. > is the centered Gaussian measure on such that for any >\W>> we have |^>>|)>=expw>,w\|)>\\=exp|i>>|)>|\<\|\|\>>|2>|)>> where >:W>\H> is the dual operator of . If \> is a measurable non-linear functional we denote by u:W\H> the following linear operator\ <\equation*> \u=\\u,h\\lim\0>h|)>-u|\>. The operator > is called Malliavin derivative and it is possible to prove that it is closable (with unique closure) on the set of measurable |)>> functions. We denote the domain of > in |)>> by >. The previous operation can be extended for functional :W\Hk>> where :W\Hk+1>> with its natural topology. Also this extension of the operator > is closable. If the measurable non-linear operator H>, where \L|)>>, is such that F,\u\|]>=\u|]>> for some \L|)>>, we say that is in the domain of the operator > and we denote by =\L|)>> the Skorokhod integral of the measurable operator . The following expression for > used in the following holds: suppose that \i>>|)>> and that F> is a trace class operator on for > almost every W> then <\equation> \=\i,-1>|)>,w\-TrF|)>. We introduce a definition for studying the random transformations defined on abstract Wiener spaces.\ <\definition> Let H> be a measurable map. We say that is a > map if for > almost every W> the map :H\H>, defined as U\U>, is a Fréchet differentiable function in and if U:H\H2>>, defined as \U\\U> where > is the Malliavin derivative, is continuous for almost every W> and with respect to the natural (Hilbert-Schmidt) topology of 2>>. We introduce the shift W> associated with , i.e. the map defined as =w+U>, and the non-linear functional :W\\> as follows <\equation> \=det+\U|)>exp-|\|>|)>, where +K|)>> is the regularized Fredholm determinant (see Chapter 9) that it is well defined for any Hilbert-Schmidt operator . In particular if is self adjoint we have\ <\equation*> det=\>|)>e>, where > are the eigenvalues of the operator . Suppose that \i>> and that U> is a trace class operator for almost any W>, then using the expression and the properties of > we obtain <\equation> \=det+\U|)>expi,-1>|)>,w\-|\|>|)>, where +K|)>> is the standard Fredholm determinant. The functional > is closely related to the transformation of the measure > with respect to the transformation . Indeed suppose that is finite dimensional then we have <\equation*> \\=exp|w,w|\>|)>x|Z>=exp|i,-1>,w|\>|)>x|Z>, where \> is a suitable normalization constant and x> is the Lebesgue measure on . Thus, if is a diffeomorphism on , we evidently have, thanks to equation , <\multline*> T>|)>|\\>=U|)>expi,-1>|)>,w\+>|\>|\>>>|||-\i,-1>|)>,U\|)>|\|>=|\|>.>>>> The previous relation can be extended to the case where and are infinite dimensional and the transformation is not a diffeomorphism but it is only a > map.\ First of all we need the following generalization to the abstract Wiener space context of the finite dimensional Sard Lemma. <\proposition> Let =w+U> be a > map and let \> be the set of the zeros of U|)>>, then the > measure of the set > is zero, i.e. |)>=0>. <\proof> See Theorem4.4.1 . The following is the change of variable theorem for (generally not invertible) > maps. \ <\theorem> Let =w+U> be an > map and let be two positive measurable functions then <\equation> >f\Tg|\|>\\=>fT>g|)>\\. In particular if W> is a measurable subset where |T|\|>> is invertible we\ <\equation*> f\T|\|>\\=>f\\. <\proof> See Theorem4.4.1 . In order to prove Theorem, and so the relationship between the weak solutions to equation and the integrals with respect to the signed measure \\>, it is not enough to consider Theorem but we need a relationship analogous to with |\|>> replaced by >. In order to achieve this result we need some more hypotheses on the map : <\description> DEG1>The map H\W> is a Fréchet differentiable map from into itself and furthermore it is > with respect to the usual topology of ; DEG2>The map is proper (i.e. inverse images of compact subsets are compact) and the equation =w> has a finite number of solution for > almost every W>.\ Under the HypothesisDEG1 and DEG2 we can define the following number <\equation*> DEG\T>sign+\U|)>|)>. This index is a suitable modification of the Leray-Schauder degree of a Fredholm non-linear operator described, for example, in Section 5.3C, where the following definition is given: if is a bounded set of such that \\B=\> and T0> for T\B> we have <\equation*> DEG=T\B>T>>|)>>. It is evident that under the HypothesisDEG2 and, as a consequence of Proposition, we have <\equation*> limW>DEG=DEG for almost all W>. <\theorem> Under the Hypotheses DEG1 and DEG2 we have that > is > almost surely equal to the constant \\> independent of and for any bounded function such that T\\\L|)>> we have <\equation*> f\T\\\=DEG\f\\. <\proof> The proof can be found in Theorem9.4.1 and Theorem9.4.6. In general is not simple to compute > but this computation simplified under the following Hypothesis: <\description> DEG3>The map >=w+\U> has bounded level set uniformly in \>, i.e. if W> is bounded \>T>> is a bounded set in . <\theorem> Under the Hypotheses DEG1, DEG2 and DEG3 we have that, for any \>: <\equation*> DEG=DEG=DEG>|)>=1. <\proof> The proposition follows from the invariance of > under homotopies of the operator . In other words for any such that >\\B=\> we have >|)>=DEG>. Under the HypothesisDEG3 we can choose big enough such that >|)>=DEG>|)>> for any\>. Since |)>=DEG|)>=1> the thesis follows. In this appendix we introduce the notion of fermionic fields used in Section and Section . \; We consider a quantum probability space ,\|)>>, where > is a separable Hilbert space and > is a positive trace class operator. If |)>> (where |)>> is the Hilbert space of bounded operators defined on >) we define |K|\>=Tr\|)>>.\ Let be a Hilbert space, we consider two continuous linear maps ,|\>:H\|)>> such that for any ,f\H> we have\ <\equation*> |)>,\|)>|}>=|\>|)>,|\>|)>|}>=|)>,|\>|)>|}>=0 where ,K|}>=K\K+K\K> is the anticommutator of operator ,K\|)>>. <\definition> Using the previous notations, the two antisymmetric fields ,|\>:H\|)>> are called fermionic fields associated with the Hilbert space if we have that\ <\equation> ||\>|)>\|)>\|\>|)>\|)>|\>=det|f,g|\>|)>. The following theorem ensure the existence of a pair of fermionic fields for each separable Hilbert space . <\theorem> For any separable Hilbert space there exists a quantum probability space ,\|)>> and two continuous linear maps ,|\>:H\|)>> such that ,|\>> are a pair of fermionic fields associated with . Furthermore, we have\ <\equation> |\|\<\|\|\>>|)>>,||\>|\<\|\|\>>|)>>\2|f|\<\|\|\>>. (we use the notation |\|\<\|\|\>>>> for the norm in a Hilbert space >). <\proof> By standard results of quantum fields theory (see, e.g., Chapter2), there are four operators >,b,b>:H\|)>> (formed by two independent pairs of anticommuting creation and anticommuting adjoint annihilation >,b>> operators) \ such that <\eqnarray> |,a|}>=,b|}>=0>|>||,b|}>=>,b|}>=0>|>||>,a|}>=>,b|}>=|f,g|\>I>,>|>>> and such that\ <\equation*> |aK|\>=|K a>|\>=|bK|\>=|K b>|\>=0 for any H> and any bounded operator |)>>. Consider now\ <\equation*> \=a>+b,|\>=b>-a, where H>. We want to prove that ,|\>> are the two fermionic fields associated with fields of the thesis of the theorem. Obviously ,|\>|}>=,\|}>=|\>,|\>|}>=0>, so we have only to prove that ,|\>> satisfy equality and inequality .\ We prove equality by induction on . By the properties of >,b,b>> we have\ <\align*> ||\>|)>\|)>|\>=>||b>|)>a>|)>|\>+|b>|)>b|)>|\>-|a|)>a>|)>|\>+>>|||a|)>b|)>|\>=|f,g|\>.>>>> Suppose that ||\>|)>\|)>\|\>|)>\|)>|\>=detf,g\|)>> we want to prove the same equality for operators. We have\ <\multline*> ||\>|)>\|)>\|\>|)>\|)>|\>=|b>|)>\|)>\|\>|)>\|)>|\>=>>||b>|)>b|)>|\>||\>|)>\|)>\|\>|)>\|)>\|\>|)>\|)>|\>>>||f,g|\>det|f,g|\>\|1,jk>|)>=det|f,g|\>|)>>>>> where we use the commutation relations of >> with >>, the induction hypothesis and the properties of determinant. Since <\equation*> |a|\<\|\|\>>|)>>=|a>|\<\|\|\>>|)>>=|b|\<\|\|\>>|)>>=|b>|\<\|\|\>>|)>>=|f|\<\|\|\>>, ,|\>> satisfy inequality. Suppose that C|)>> for some continuous injection , then by the identification of with its dual we have that >|)>\H>, where \|)>|)>>> is the Dirac delta with mass in \>. In this way we can define ,|\>> as continuous function of the point > in the following way <\equation*> \:=\>|)>|)>|\>:=|\>>|)>|)> and the corresponding covariance function as <\equation*> S|)>=||\>|)>\|\>. Hereafter we suppose that |)>> is a continuous function of the form |)>=S|)>\0>. In this case, if L|)>>, by Theorem we have |\|\>|\<\|\|\>>|)>>\2S> and thus >g|\>\\x> is a bounded well defined operator. Under the previous condition the operator :L|)>\L|)>>, defined as =gS|)>h|)>\x>, is trace class since\ <\equation*> Tr|\|>|)>\>|\|>Tr|\>\\|\|>|)>\x\2S|g|\<\|\|\>>|)>>\+\. This means that the Fredholm determinant (see Chapter 3) |)>> is well defined and finite. Furthermore, we have the following representation. <\theorem> Under the previous hypotheses and notations we have <\equation*> |exp>g|\>\\x|)>|\>=det|)>. <\proof> By Definition and the definition of the function , we have that\ <\multline*> |>g|\>\\x|)>|\>=>>|>g|)>\g|)>det-x|)>|)>\x\\x=>>|>det|)>S-x|)>>|>||)>S-x|)>>>|>|>|>>||)>S-x|)>>|>||)>S-x|)>>>>>>|)>\x\\x.>>>> On the other hand, when is continuous, by Theorem3.10 of, we have that\ <\multline> |)>=>>|>>det|)>S-x|)>>|>||)>S-x|)>>>|>|>|>>||)>S-x|)>>|>||)>S-x|)>>>>>>|)>\x\\x.>>>> The thesis follows. <\remark> The fermionic fields considered in Section and Section , where \>|)>>, ,2>|)>> with norm |f|\<\|\|\>>=+m|)>>f\x>, satisfies all the hypotheses of Theorem . <\bibliography|bib|tm-plain|elliptic-stoch-quant2> <\bib-list|60> S.Albeverio, F.C.De VecchiS.Ugolini. Entropy chaos and Bose-Einstein condensation. , 168(3):483\U507, 2017. S.Albeverio, H.KawabiM.Röckner. Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions. , 262(2):602\U638, jan 2012. S.AlbeverioS.Kusuoka. The invariant measure and the flow associated to the >-quantum field model. , nov 2017. To appear in Ann. Scuola Normale di Pisa. S.AlbeverioM.Röckner. Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. , 89(3):347\U386, 1991. S.AlbeverioM.W Yoshida. > maps and elliptic spdes with polynomial and exponential perturbations of Nelson's euclidean free field. , 196(2):265\U322, 2002. C.D.AliprantisKimC.Border. . Springer, Berlin, Third, 2006. A hitchhiker's guide. A.Arai. Supersymmetric extension of quantum scalar field theories. , 16, 73\U90. Kluwer Acad. Publ., Dordrecht, 1993. JohnC.Baez, IrvingE.SegalZheng-Fang Zhou. . Princeton Series in Physics. Princeton University Press, Princeton, NJ, 1992. H.Bahouri, J.-Y.CheminR.Danchin. , 343. Springer, Heidelberg, 2011. G.Benfatto, G.GallavottiF.Nicolò. Elliptic equations and Gaussian processes. , 36(3):343\U400, 1980. M.S.Berger. . Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis, Pure and Applied Mathematics. V.S.Borkar, R.T.ChariS.K.Mitter. Stochastic quantization of field theory in finite and infinite volume. , 81(1):184\U206, nov 1988. D.C.BrydgesJ.Z.Imbrie. Dimensional Reduction Formulas for Branched Polymer Correlation Functions. , 110(3):503\U518, mar 2003. D.BrydgesJ.Imbrie. Branched polymers and dimensional reduction. , 158(3):1019\U1039, nov 2003. G.Da PratoA.Debussche. Strong solutions to the stochastic quantization equations. , 31(4):1900\U1916, 2003. G.Da PratoJ.Zabczyk. . Cambridge University Press, feb 2008. P.H.DamgaardH.Hüffel. Stochastic quantization. , 152(5-6):227\U398, aug 1987. P.H.DamgaardH.Hüffel. . World Scientific, 1988. G.DaPrato. . Birkhäuser, Basel ; Boston, 2004 edition, feb 2005. A.deGoursac. Noncommutative supergeometry and quantum supergroups. 597, 12028. Eprint: arXiv:1501.06316, apr 2015. B.DeWitt. . Cambridge University Press, Cambridge ; New York, 2 edition, may 1992. L.C.Evans. , 19. American Mathematical Society, Providence, RI, 1998. A.L.FetterJ.D.Walecka. . Dover Books on Physics. Dover Publications, 2012. David GilbargNeilS.Trudinger. . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. James GlimmArthur Jaffe. . Springer-Verlag, New York, Second, 1987. A functional integral point of view. L.Gross. Abstract Wiener spaces. , 31\U42. Univ. California Press, Berkeley, Calif., 1967. M.GubinelliM.Hofmanová. Global solutions to elliptic and parabolic > models in Euclidean space. , apr 2018. M.HairerK.Matetski. Discretisations of rough stochastic PDEs. , 46(3):1651\U1709, may 2018. Martin HairerMassimo Iberti. Tightness of the Ising-Kac model on the two-dimensional torus. , 171(4):632\U655, 2018. T.Helmuth. Dimensional Reduction for Generalized Continuum Polymers. , 165(1):24\U43, oct 2016. Y.ImryS.-K.Ma. Random-Field Instability of the Ordered State of Continuous Symmetry. , 35:1399\U1401, nov 1975. K.Iwata. An infinite dimensional stochastic differential equation with state space >. , 74(1):141\U159, mar 1987. G.Jona-LasinioP.K.Mitter. On the stochastic quantization of field theory. , 101(3):409\U436, 1985. R.KhasminskiiG.N.Milstein. . Springer, Heidelberg; New York, 2nd ed. 2012 edition, sep 2011. A.Klein. Supersymmetry and a two-dimensional reduction in random phenomena. , 1136, 306\U317. Springer, Berlin, 1985. A.Klein, L.J.LandauJ.F.Perez. Supersymmetry and the Parisi-Sourlas dimensional reduction: a rigorous proof. , 94(4):459\U482, 1984. A.KleinJ.F.Perez. Supersymmetry and dimensional reduction: A non-perturbative proof. , 125(6):473\U475, jun 1983. A.KleinJ.F.Perez. On the density of states for random potentials in the presence of a uniform magnetic field. , 251:199\U211, jan 1985. LauraM.MoratoStefania Ugolini. Stochastic description of a Bose-Einstein condensate. , 12(8):1601\U1612, 2011. J.-C.MourratH.Weber. Global well-posedness of the dynamic > model in the plane. , 45(4):2398\U2476, jul 2017. J.Zabczyk,G.DaPrato N.V. Krylov, M. Röckner. . Lecture Notes in Mathematics 1715. Springer, 1, 1999. E.Nelson. Derivation of the Schrödinger equation from Newtonian mechanics. , (150):1079\U1085, 1966. E.Nelson. . Princeton University Press, Princeton, N.J., 1967. E.Nelson. The free markoff field. , 12(2):211\U227, 1973. D.Nualart. . Probability and its Applications (New York). Springer-Verlag, Berlin, Second, 2006. G.ParisiN.Sourlas. Random Magnetic Fields, Supersymmetry, and Negative Dimensions. , 43(11):744\U745, sep 1979. G.ParisiN.Sourlas. Supersymmetric field theories and stochastic differential equations. , 206(2):321\U332, oct 1982. G.ParisiY.S.Wu. Perturbation theory without gauge fixing. , 24(4):483\U496, 1981. A.Rogers. . World Scientific, Singapore; Hackensack, NJ, 2007. OCLC: 648316841. W.Rudin. . McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. Albert SchwarzOleg Zaboronsky. Supersymmetry and localization. , 183(2):463\U476, 1997. B.Simon. |)>> Euclidean (quantum) field theory>. Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. B.Simon. , 120. American Mathematical Society, Providence, RI, Second, 2005. W.Stannat. Stochastic partial differential equations: Kolmogorov operators and invariant measures. , 113(2):81\U109, 2011. Hans Triebel. , 78. Birkhäuser Verlag, Basel, 1983. A.S.ÜstünelM.Zakai. . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. J.van derHoeven. . Citeseer, 2003. Franz Wegner. , 920. Springer, Heidelberg, 2016. Grassmann variables and the method of supersymmetry. A.P.Young. On the lowering of dimensionality in phase transitions with random fields. , 10(9):0, may 1977. J.Zinn-Justin. , 85. The Clarendon Press, Oxford University Press, New York, Second, 1993. Oxford Science Publications. <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> damgaard_stochastic_1987 damgaard_stochastic_1988 parisi_perturbation_1981 nelson1966 MR0214150 Nelson1973 parisi_perturbation_1981 parisi_perturbation_1981 da_prato_stochastic_2008 krylov_stochastic_1999 prato_kolmogorov_2005 khasminskii_stochastic_2011 Albeverio2017 MR2768734 Ugolini2011 albeverio_stochastic_1991 Albeverio2002 albeverio_invariant_2017 Gallavotti1980 borkar_stochastic_1988 da_prato_strong_2003 GH18 hairer_discretisations_2018 hairer_tightness_2018 jona-lasinio_stochastic_1985 MW17 albeverio_strong_2012 iwata_infinite_1987 GH18 imry_random_field_1975 young_lowering_1977 parisi_random_1979 parisi_supersymmetric_1982 parisi_random_1979 parisi_supersymmetric_1982 parisi_supersymmetric_1982 damgaard_stochastic_1988 parisi_supersymmetric_1982 klein_supersymmetry_1985 Klein1984 klein_supersymmetry_1983 klein_density_1985 brydges_branched_2003 brydges_dimensional_2003 helmuth_dimensional_2016 iwata_infinite_1987 albeverio_strong_2012 Klein1984 parisi_supersymmetric_1982 Klein1984 Ustunel2000 Klein1984 Gross1967 Nualart2006 Ustunel2000 Bahouri2011 Trie1983 Gilbarg2001 Evans1998 Aliprantis2006 Ustunel2000 Simon2005 Simon2005 Ustunel2000 Ustunel2000 Aliprantis2006 Rudin1973 parisi_random_1979 Klein1984 Aliprantis2006 Nualart2006 Ustunel2000 Simon2005 Simon2005 Nualart2006 Van2003majorants Klein1984 Klein1984 Ustunel2000 Klein1984 Simon1974 Glimm1987 Fetter2012 dewitt_supermanifolds_1992 arai_supersymmetric_1993 rogers_supermanifolds_2007 de_goursac_noncommutative_2015 Klein1984 Wegner2016 Zinn1993 Klein1984 Zaboronsky1997 Fetter2012 Klein1984 Klein1984 Klein1984 Klein1984 Klein1984 Ustunel2000 Klein1984 Klein1984 Ustunel2000 Simon2005 Ustunel2000 Ustunel2000 Berger1977 Ustunel2000 Segal1992 Simon2005 Simon2005 <\associate|toc> |math-font-series||1Introduction> |.>>>>|> |math-font-series||2The elliptic SPDE> |.>>>>|> |2.1Strong solutions |.>>>>|> > |2.2Weak solutions |.>>>>|> > |math-font-series||3Elliptic stochastic quantization> |.>>>>|> |math-font-series||4Dimensional reduction> |.>>>>|> |4.1Bounded potentials |.>>>>|> > |4.2Potentials satisfying Hypothesis |V>> and |C> |.>>>>|> > |4.3Potentials satisfying only Hypothesis |V>> |.>>>>|> > |math-font-series||5Supersymmetry> |.>>>>|> |5.1The superspace |.>>>>|> > |5.2The superfield |.>>>>|> > |5.3The supersymmetry |.>>>>|> > |5.4Expectation of supersymmetric polynomials |.>>>>|> > |math-font-series||6Removal of the spatial cut-off> |.>>>>|> |math-font-series||Appendix ATransformations in abstract Wiener spaces> |.>>>>|> |math-font-series||Appendix BFermionic fields > |.>>>>|> |math-font-series||Bibliography> |.>>>>|>