1Introduction

One of the first steps in constructive quantum field theory (QFT) is to build a family of distributions satisfying Wightman axioms [GJ87, SW64], which can thus be interpreted as the Wightman functions of a field theory with unique ground state, invariant with respect to the Poincaré group. Wightman axioms, however, do not identify uniquely a particular QFT. Instead different QFTs are expected to satisfy different Dyson–Schwinger equations: a systems of partial differential equations (PDEs) relating Wightman functions and encoding the local and hyperbolic equations of motions for the quantum fields [dM97, Sch51].

An important progress in the analysis of Wightman QFTs was the introduction of Schwinger functions, namely the analytic continuations on imaginary time of Wightman functions, which are described by a set of axioms introduced by Osterwalder and Schrader (see [GJ87, OS73, OS75, Sim74]). As observed by Nelson (see [Nel73a, Nel73b, Nel73c]), in many cases (such as the scalar bosonic QFT) Schwinger functions are the moments of a probability measure \(\nu\) on Schwarz distributions (see [GJ87, Sim74] for systematic applications of this approach). In particular, the Dyson–Schwinger equations translate in an integration by parts (IbP) formula for the measure \(\nu\) [ALZ06, DG74, FR77, GH21] and it becomes natural to investigate the problem of existence and uniqueness of probability measures satisfying prescribed integration by parts formulas.

The characterization of a measure through some IbP formula is a classical subject in stochastic analysis which has different formulations, such as existence and uniqueness of a measure with given logarithmic gradient [Bog10] or the unique closability of a minimally defined pre-Dirichlet form [ABR22, ARZ93, BR95] (and the references therein). The application of logarithmic gradients and integration by parts formulas to quantum field theory was already proposed by Kirillov in the case of sine–Gordon models, see [Kir94a, Kir94b, Kir95a, Kir95b] where the problem is considered without renormalization. In the aforementioned works by Kirillov, the author exploits also a Lyapunov functions technique to show existence of solutions to the integration by parts formulas; a generalization of this technique is applied to the related problem of non-singular (i.e. with no need for renormalization) Fokker–Planck–Kolmogorov equations by Bogachev and Röckner [BR99, BR01]. The problem of uniqueness of solutions to integration by parts formula or, similarly, to an infinite-dimensional Fokker–Planck–Kolmogorov equation, is solved only in some particular cases, see for instance the books [Bog10, BKRS15], and the works by Bogachev, Da Prato and Röckner [BDR09, BDR11] and by Röckner, Zhu and Zhu [RZZ14], where a dissipative non-regular drift (without renormalization) is considered. Let us also mention the studies about uniqueness of solutions to Fokker–Planck–Kolmogorov equations or of invariant measures of the \(P (\varphi)_2\) stochastic quantization equation on the two-dimensional torus [DD20, KT22, RZZ17a, RZZ17b, TW18]. Although in this case the problem of existence and uniqueness is solved, it is not clear whether the techniques used in the aforementioned papers can be extended to the models on the non-compact space \(\mathbb{R}^2\) or in dimension greater than two. It is worth to mention that the study of uniqueness in the framework of Dirichlet forms for the exponential model has been discussed by Albeverio, Kawabi and Röckner in [AKR12] for the one-dimensional non-singular case, and by the same authors together with Mihalache in [AKMR21] for the two-dimensional setting on the torus. See also [AMR15] for a review of the existing literature on the Dirichlet form approach to the dynamical problem. Let us mention that our concern here is different from that in Dirichlet form theory since here the measure is an unknown in the problem and not given a priori.

The key open problem that we address in this paper is to provide a suitable setting in which existence and uniqueness of measures satisfying given some singular IbP formula, that is one involving a renormalization procedure in its definition, which is the usual situation in constructive Euclidean QFT. Instead of attempting a general framework, we concentrate in a particular case where we can establish a reasonable well-posedness theory for the singular IbP problem: we test our ideas on the EQFT with exponential interaction and positive mass \(m > 0\), or Høegh–Krohn model [AH74] on the whole space \(\mathbb{R}^2\). The exponential interaction in the case of mass \(m = 0\) [AGH79, AJPS97, RV14] is a classical example of conformal field theory [Mus10, Sch08] and it finds important applications in Liouville quantum gravity [DS11, KRV20]. As far as we know, this contribution of ours is the first which manages to address this question for an EQFT requiring renormalization and in the infinite volume limit.

Let us give a more detailed description of the problem that we consider here. Let \(\mathcal{S} (\mathbb{R}^2)\) be the space of Schwartz functions and denote its dual, that is the space of tempered distributions, by \(\mathcal{S}' (\mathbb{R}^2)\). We fix a Banach space (or a topological vector space) \(E \subset \mathcal{S}' (\mathbb{R}^2)\) and we consider a family \(\mathfrak{F}\) of functions \(F : E \rightarrow \mathbb{R}\) which are Fréchet differentiable. In particular, for any \(\varphi \in E\), we can consider the derivative \(D_f F (\varphi)\) of \(F\) in the direction \(f \in \mathcal{S} (\mathbb{R}^2)\); the map \(f \mapsto D_f F (\varphi)\) is linear and bounded in \(E\), and thus in \(\mathcal{S} (\mathbb{R}^2)\). This means that, since \(\mathcal{S}' (\mathbb{R}^2)\) is the topological dual of \(\mathcal{S} (\mathbb{R}^2)\), there exists a unique \(\nabla_{\varphi} F (\varphi) \in \mathcal{S}' (\mathbb{R}^2)\) such that \(D_f F (\varphi) = \langle \nabla_{\varphi} F (\varphi), f \rangle\), where \(\langle \cdot, \cdot \rangle = \langle \cdot, \cdot \rangle_{\mathcal{S}', \mathcal{S}}\) is the duality between \(\mathcal{S}' (\mathbb{R}^2)\) and \(\mathcal{S} (\mathbb{R}^2)\).

Let us denote by \(\mathcal{M}\) a family of probability measures on \(E\), and let \(B : E \rightarrow \mathcal{S}' (\mathbb{R}^2)\) a given map. Then a generic IbP problem, or Euclidean Schwinger–Dyson equation for a measure \(\nu \in \mathcal{M}\), has the general form

We are interested in local functionals \(B\) which have the form

for some smooth function \(p : \mathbb{R} \rightarrow \mathbb{R}\). This locality of the IbP formula is peculiar of EQFT where locality (or reflection positivity, or domain Markov property) is structurally linked to the finite speed of propagation of signals in the Minkowski theory. Unfortunately, such kind of functions \(B\) are seldom well-defined on the set \(E\) on which we could hope that any solution \(\nu\) would be supported. Typically, this support looks very much like the support of the GFF and therefore non-linear local functionals are not automatically well-defined and need to be approached via an ultraviolet regularization and subsequent renormalization. In this sense, we talk about a singular IbP formula adopting the term from the recent literature in singular stochastic PDEs.

Given this motivation we consider a sequence of maps \((B_{\varepsilon})_{\varepsilon > 0}\) such that, for every \(\varepsilon > 0\), we have \(B_{\varepsilon} : E \rightarrow \mathcal{S}' (\mathbb{R}^2)\) and for which we recover locality in the limit as \(\varepsilon \rightarrow 0\). They are typically of the form

where \((g_{\varepsilon})_{\varepsilon \geqslant 0}\) is some sequence of local smoothing kernels for which \((g_{\varepsilon} \ast \varphi) \rightarrow \varphi\) in \(\mathcal{S}' (\mathbb{R}^d)\) and \((p_{\varepsilon} : \mathbb{R} \rightarrow \mathbb{R})_{\varepsilon}\) is a sequence of smooth function chosen to deliver the expected renormalization, typical of EQFT in two and three dimensions. The main problem we analyze in this paper can therefore be formulated abstractly as follows:

Problem A. We say that a measure \(\nu \in \mathcal{M}\) satisfies the integration by parts formula with respect to \((B_{\varepsilon})_{\varepsilon > 0}\) and \(\mathcal{M}\) if, for any \(f \in \mathcal{S} (\mathbb{R}^2)\), we have

where \(\operatorname{Cyl}_E^b\) is the set of smooth and bounded cylinder functions (cf. the Notation section at the end of the present introduction).

Let us remark that the problem of existence and uniqueness of solutions to Problem A strongly depends on the subset \(\mathcal{M}\) of the space \(\mathcal{P} (E)\) of (Radon) probability measures on \(E\). One of the main problems is that, if we consider \(\mathcal{M}=\mathcal{P} (E)\), i.e. we consider a generic Radon probability measure on \(E\), then it is not clear if \((B_{\varepsilon})_{\varepsilon}\) admits a limit in probability as \(\varepsilon \rightarrow 0\) and whether the limit depends on the measure \(\nu\) or not. For example, if we take \(B_{\varepsilon}\) to be the drift of \(\Phi^4_2\) measure in \(\mathbb{R}^2\), namely

with \(c_{\varepsilon} = \| (- \Delta + m^2)^{- 1 / 2} g_{\varepsilon} \|^2_{L^2 (\mathbb{R}^2)}\), then it is known that \(B_{\varepsilon}\) converges to the unique limit \(: \varphi^4 :\)(where \(: \cdot :\) is the Wick product of Gaussian random fields, see Chapter 1 in [Sim74]) when the measure \(\nu\) is absolutely continuous with respect to the Gaussian free field with mass \(m > 0\). On the other hand, if \(\nu\) is supported on the space of smooth functions, such a limit does not exist. It is useful then to consider a class of measures \(\nu\) for which it is possible to make sense (almost surely) of the limit \(\lim_{\varepsilon \rightarrow 0} B_{\varepsilon}\).

The set \(\mathcal{M}\) can therefore be neither too large nor too small and we find useful to focus on measures which “look like” the Gaussian free field with mass \(m > 0\). We formalize this idea introducing an appropriate Wasserstein metric which encodes this proximity without requiring absolute continuity, which in the full \(\mathcal{S}' (\mathbb{R}^2)\) is anyway never an appropriate condition. Let \(B\) be a convex cone in \(E\) equipped with a norm \(\| \cdot \|_B\) that is stronger than the one of \(E\), then we define the function \(d_B : \mathcal{P} (E) \times \mathcal{P} (E) \rightarrow [0, + \infty]\) as

where \(\Pi (\nu, \nu') \subset \mathcal{P} (E^2)\) is the set of couplings between \(\nu\) and \(\nu'\), and

Let us suppose that \(E\) contains the support of the measure \(\nu^{\operatorname{free}}\) of the Gaussian free field with mass \(m > 0\) and define

The space \(\mathcal{M}_B\) strongly depends on the convex cone \(B\) together with its norm. For example, if we consider \(B = H^1 (\mathbb{R}^2)\) we have

Another simple case is when \(B = E\) equipped with its natural norm, this gives

namely the Wasserstein space of measures on \(E\) (see Chapter 6 in [Vil09]).

The class of measures \(\mathcal{M}_B\) encodes the existence of sufficiently regular couplings between our target measures and the GFF. This mirrors the situation in singular SPDEs and in other recent development in EQFT where the existence of such couplings has been a key technical aspect to develop a suitable stochastic analysis of singular dynamics and EQFTs. In the fundamental work [DD03], Da Prato and Debussche indeed introduced a notion of solution to stochastic quantization equations in two dimensions as the sum of the Gaussian free field with mass \(m\) and a solution to a (random) non-linear PDE. In other words, they look at the solution as a perturbation of the solution to the linear stochastic heat equation. More recently, Barashkov and Gubinelli [BG20, BG21a, BG21b] developed a coupling approach based on an optimal control problem, Bauerschmidt and Hofstetter [BH20] used a similar coupling to study the pathwise properties of the two dimensional sine–Gordon EQFT on a torus and Shenfeld and Mikulincer [She22, MS21] linked these ideas with current developments in the theory of optimal transport and functional inequalities. Similar ideas are fundamental also in the context of singular stochastic PDEs, see regularity structures [CCHS22a, CCHS22b, Hai14, HS16], paracontrolled calculus [AK20, AK21, GH21, GIP15], and renormalization group theory [Duc21, Kup16].

As we already mentioned, we would like to provide complete well-posedness results within this framework of singular IbP formulas and, for this reason, we focus on the specific case of the exponential interaction, namely we take \(E = B_X + B_Y\), where \(B_X = C^{- \delta}_{\ell} (\mathbb{R}^2)\), i.e. the (weighted) Besov–Hölder space with negative regularity \(- \delta\) (see Appendix A), and \(B_Y\) is taken to be

where \(s (\alpha) > 0\) satisfies some conditions depending on the parameter \(\alpha\) (see Definition 2.6 below) and \(B^{s (\alpha) - \delta}_{p, p, \ell}\) is a weighted Besov space (see Appendix A). Moreover, we set

where \(\alpha, m \in \mathbb{R}_+\), \(f_{\varepsilon}\) is a smooth, spatial cut-off function such that \(f_{\varepsilon} \rightarrow 1\), \(g_{\varepsilon} = \varepsilon^{- 2} g (\varepsilon^{- 1} \cdot)\) is a regular mollifier, and

is a renormalization constant diverging logarithmically to \(+ \infty\) as \(\varepsilon \rightarrow 0\). Finally, we consider the space of measures \(\mathcal{M}\) in Problem A to be \(\mathcal{M}_{B_Y}\) (see equation (1.2)), that is an intermediate regime between the case \(\mathcal{M}_{H^1}\) (i.e. the space of measures that are absolutely continuous with respect to the Gaussian free field) and \(\mathcal{M}_E\) (which coincides with the Wasserstein space \(\mathcal{W}^1_E\)).

Remark 1.1. In the following, we shall consider the number \(\gamma_{\max} \approx 0.55\) given by the maximum taken over all \(r > 1\), satisfying Definition 2.6, of

Moreover, we let \(\tilde{\gamma}_{\max} := 3 - 2 \sqrt{2} \approx 0.172 < \gamma_{\max}\).

In the present setting, it is possible to obtain the following results.

It is also possible to obtain an existence result for the whole regime \(\alpha^2 < 8 \pi\).

Theorem 1.3 is obtained by building suitable Lyapunov functions independent of \(\varepsilon > 0\), similarly to what was done by Kirillov for the not-renormalized equation (see [Kir94a, Kir94b, Kir95a, Kir95b]), and reducing the infinite-dimensional problem to the existence of a symmetric invariant measure for a finite-dimensional differential operator.

An important consequence of Theorem 1.2 and Theorem 1.3 is a differential characterization of the exponential measure (see Theorem 2.13 for details).

Let us mention that in the present paper we also get a uniqueness result for a slightly more restrictive formulation of Problem A (cf. Problem B in Section 2.1) in the regime \(\alpha^2 < 4 \pi \gamma_{\max}\). The possible measures solving this latter formulation of the IbP problem contain the invariant measure of the stochastic quantization equation with exponential interaction.

Plan of the paper

Let us present here the structure of the paper. In Section 2, we discuss the IbP Problem A in the general setting, establishing some equivalent formulations that will be useful to address Problem A itself. We also consider the more restrictive Problem B and Problem B-sym, their relation with Problem A and some properties of the solutions to such problems, such as the negativity of the coupling between the Gaussian free field and the quantum field measure with exponential interaction. Section 3 is devoted to the study of uniqueness of solutions to Problem A showing the uniqueness result stated in Theorem 1.2. Since the proof relies on some properties of the solution to the resolvent equation associated to the drift \(B_{\varepsilon}\), most parts of the section is dedicated to the study of such an object. The existence of solutions to Problem A is proved in Section 4. The proof is based on an approximation method and it involves Lyapunov functions. In Appendices A and B we recall the definitions and properties of weighted Besov spaces and Wick exponential, respectively, which are used throughout the paper. Appendices C, D, and E are concerned with some technical, analytical results on (S)PDEs exploited in the paper.

Notation

We fix here some notation adopted throughout the paper.

We write \(a \lesssim b\) or \(b \gtrsim a\) if there exists a constant \(C > 0\), independent of the variables under consideration, such that \(a \leqslant C b\), and \(a \simeq b\) if both \(a \lesssim b\) and \(b \lesssim a\). If the aforementioned constant \(C\) depends on a variable, say \(C = C (x\)), then we use the notation \(a \lesssim_x b\), and similarly for ≳.

The space \(L (A, B)\) is the space of linear and bounded functionals from the Banach space \(A\) into the Banach space \(B\). \(L (A, B)\) is equipped with its natural operator norm \(\| \cdot \|_{L (A, B)}\). We also write \(L (A) = L (A, A)\).

The space of Schwartz functions on \(\mathbb{R}^d\) is denoted by \(\mathcal{S} (\mathbb{R}^d)\) and its dual, that is the space of tempered distributions, is denoted by \(\mathcal{S}' (\mathbb{R}^d)\). Moreover, we let \(C_b^m (\mathbb{R}^d)\) be the space of \(m\)-times differentiable functions on \(\mathbb{R}^d\) with continuous and bounded derivatives. We also use the notation \(B^r_{p, q, \ell} (\mathbb{R}^d)\) to denote the weighted Besov space on \(\mathbb{R}^d\) (see Appendix A for more results on Besov spaces). For \(k, \ell > 0\), we define the weight \(\rho^k_{\ell} (x) = (1 + k | x |^2)^{- \ell / 2}\), \(x \in \mathbb{R}^d\), and \(\rho_{\ell} := \rho^1_{\ell}\).

Let \(K\) be a convex subset of \(\mathcal{S}' (\mathbb{R}^d)\), we define the set \(\operatorname{Cyl}_K\) of cylinder functions on \(K\) to be the set of functions \(F : K \rightarrow \mathbb{R}\) such that there exist a function \(\tilde{F} \in C^2 (\mathbb{R}^n)\), with at most linear growth at infinity and all bounded derivatives, and \(u_1, \ldots, u_n \in \mathcal{S} (\mathbb{R}^d)\), such that \(F (\kappa) = \tilde{F} (\langle u_1, \kappa \rangle, \ldots, \langle u_n, \kappa \rangle)\), for every \(\kappa \in K\). We say that \(F \in \operatorname{Cyl}_K\) has compact support in Fourier variables if the Fourier transforms \(\hat{u}_j\) of \(u_j\), \(j = 1, \ldots, n\), have compact supports on \(\mathbb{R}^d\). Moreover, we denote by \(\operatorname{Cyl}^b_K\) the subset of \(\operatorname{Cyl}_K\) such that, sticking with the previous notation, the function \(\tilde{F}\) is also bounded.

We adopt the notation \(P_t = e^{- (- \Delta + m^2) t}\) for the heat kernel with mass \(m\) (cf. Appendix A).

Acknowledgements

The authors thank Sergio Albeverio for the kind suggestions and remarks on various early drafts of the paper. This work has been partially funded by the German Research Foundation (DFG) under Germany's Excellence Strategy - GZ 2047/1, project-id 390685813. The first author is partially funded by the Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM), Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA): “Analisi armonica e stocastica in problemi di quantizzazione e integrazione funzionale”.

2Formulation of the problem

2.1Reformulation of Problem A

In this section, we want to reformulate Problem A when the space of measures \(\mathcal{M}\) is the space \(\mathcal{M}_B\) introduced in (1.2). For this reason, we consider \(E = B_X + B_Y\), where \(B_X\) is some space supporting the measure of the Gaussian free field with mass \(m > 0\), and \(B_Y = B\). We consider also the natural projection \(P^X : B_X \times B_Y \rightarrow B_X\) and the map \(P^{X + Y} : B_X \times B_Y \rightarrow E\) such that \((X, Y) \mapsto X + Y\).

Let us reformulate Problem A by means of a second order operator. Consider \(B_{\varepsilon}\) to be regular enough and let

Problem A'. We say that a measure \(\nu \in \mathcal{M}\) satisfies the symmetric Fokker–Planck–Kolmogorov equation related to \(B_{\varepsilon}\) if

Remark 2.1. A consequence of equation (2.1) is that

That is what is usually called Fokker-Planck-Kolmogorov equation.

We now want to lift the problem from the space \(E\) to the space \(B_X \times B_Y\). In order to do so, we introduce the following notion:

Remark 2.3. With the same notation as in Definition 2.2, it is clear that \(\mathcal{M}_{B_Y}\) defined as in (1.2) where \(B = B_Y\), coincides with the space \(P^{X + Y}_{\ast} \mathcal{M}_{B_X \times B_Y}\). Indeed, \(\nu \in \mathcal{M}_{B_Y}\) if and only if there exists a coupling \(\pi (\mathrm{d} x, \mathrm{d} z)\) between \(\nu (\mathrm{d} z)\) and the free field \(\mu^{\operatorname{free}} (\mathrm{d} x)\) such that the difference \(x - z\) is supported on \(B_Y\). The coupling \(\pi\) is related with a measure \(\mu\) in \(\mathcal{M}_{B_X \times B_Y}\) via the transformation \(y = x - z\).

We consider now an operator \(\mathcal{L}_{\varepsilon}\) on the space of regular functions on \(B_X \times B_Y\) of the form

Problem A”. We say that a measure \(\nu \in \mathcal{M}_{B_Y}\) satisfies the symmetric Fokker–Planck–Kolmogorov equation related to \(B_{\varepsilon}\) if

where \(\mu\) is a coupling of \(\nu\) with the free field.

Remark 2.4. As in the case of Problem A' (see also Remark 2.1), equation (2.3) implies

We often say that \(\mu\) solves Problem A” if there is a measure \(\nu\) solving the FPK equation with \(\mu\) as coupling of \(\nu\) with the free field.

Problem A” is based on a formulation of integration by parts formula related to stationary solutions to the Fokker–Planck–Kolmogorov equation, making the operator \(\mathcal{L}_{\varepsilon}\) symmetric on its domain (see Problem A' in Section 2.1 or [ABR99, BR95] for more details on the relations between the formulations).

It is convenient to introduce new equations related to Problem A” described above, where the argument of \(\mathcal{L}_{\varepsilon}\) in equation (2.4) is not necessarily of the form \(F \circ P^{X + Y}\).

Problem B. We say that a Radon measure \(\mu\) on \(B_X \times B_Y\) satisfies the Fokker–Planck–Kolmogorov equation related to \(B_{\varepsilon}\), if

Problem B-sym. We say that \(\mu\) satisfies the symmetric Fokker–Planck–Kolmogorov equation related to \(B_{\varepsilon}\) if, furthermore,

for any \(F, G : B_X + B_Y \rightarrow \mathbb{R}\) such that \(F \circ P^{X + Y} \in \operatorname{Cyl}_E^b, G \circ P^{X + Y} \in \operatorname{Cyl}_E\).

We have then the following result.

Proof. Let us give the proof point by point.

Proof of point i. Let us prove that a solution to Problem A' is also a solution to Problem A. Let \(\nu \in \mathcal{M}\) solve Problem A', take \(F \in \operatorname{Cyl}^b_E\), and consider \(G (\cdot) = \langle \cdot, f \rangle\), for \(f \in \mathcal{S} (\mathbb{R}^2)\). Then, by equation \(\textrm{(\hyperref[eqn:cond01]{2.2})}\), we have

where

Thus, we have

We now show that any solution to Problem A is a solution to Problem A'. Let \(\mu \in \mathcal{M}\) solve Problem A. Consider an orthonormal basis \((e_n)_n\) of \(L^2 (\mu)\) and take \(F = \langle \nabla G, e_n \rangle\), for some \(G \in \operatorname{Cyl}_E\). We have then, by equation (1.1),

Taking the sum over \(n \in \mathbb{N}\) on both sides, noticing that by the properties of cylinder functions we can exchange the integral with the sum, and exploiting Parseval's identity we have

that is \(\int \mathfrak{L}_{\varepsilon} G \mathrm{d} \nu = 0\). Now, consider again an orthonormal basis \((e_n)_n\) of \(L^2 (\mu)\), but take \(F = G \langle \nabla H, e_n \rangle\), for some \(G, H \in \operatorname{Cyl}_E\). Let us note that

On the other hand, equation (1.1) gives

This yields

and taking the sum over \(n \in \mathbb{N}\) and using again the properties of cylinder functions, we get

which implies

namely

Doing the same computation exchanging the roles of \(G\) and \(H\) we also get \(\int H\mathfrak{L}_{\varepsilon} (G) \nu (\mathrm{d} \varphi) = - \int \langle \nabla G, \nabla H \rangle \nu (\mathrm{d} \varphi)\), that gives

and concludes the proof of the first point of the theorem.

Proof of point ii. Assume further that \(\mathcal{M}=\mathcal{M}_{B_Y}\). Notice that \(\mathcal{L}_{\varepsilon} (F \circ P^{X + Y}) =\mathfrak{L}_{\varepsilon} (F) \circ P^{X + Y}\), and since \(P^{X + Y}_{\ast} \mu = \nu\) we have that

which gives the equivalence between equations (2.4) and (2.2) and in the same way we have the equivalence between equations (2.3) and (2.1).

Proof of point iii. Let \(\mu\) be a solution to Problem B. We only have to show that \(\mu \in \mathcal{M}_{B_X \times B_Y}\), that is, the only condition that we need to verify is that \(P^X_{\ast} \mu = \mu^{\operatorname{free}}\).

Since the terms involving derivatives with respect to \(X\) of \(\mathcal{L}_{\varepsilon}\) do not depend on \(\varepsilon\), in this proof we write \(\mathcal{L}\) for the operator \(\mathcal{L}_{\varepsilon}\). Furthermore, for a solution \(\mu\) to (2.5)–(2.6) we have that, if \(\Phi\) does not depend on \(Y\),

Step 1. We start with the proof that the free field measure with mass \(m > 0\) solves equations (2.5)–(2.3).

Consider, for \(X_0 \in B_X\),

and let \(\mu_{\infty}\) be the limit

We want to show that \(\int \mathcal{L} \Phi \mathrm{d} \mu_{\infty} = 0\), for any \(\Phi\) smooth cylindric function. By Itô's formula (see Theorem 4.32 in [DZ14]), taking expectation and dividing by \(T > 0\), we have that

and letting \(T \rightarrow + \infty\), recalling that \(\Phi\) is bounded, we get

Step 2. We now show uniqueness of the measure \(\mu\). Consider

where \(F \in \operatorname{Cyl}_{B_X}\) of the form

Step 2.a. Let us prove that \(\mathcal{L}_{\varepsilon} G^{\lambda}\) is well-defined, recall

where

and

since

This gives

and thus \(\mathcal{L}G^{\lambda}\) is well-defined.

Step 2.b. We now show that

By Itô formula (see Theorem 4.32 in [DZ14]), we have

and taking the expectation we get

On the other hand,

and taking the expectation

Comparing equations (2.8) and (2.9), we get

dividing by \(t > 0\) and letting \(t \rightarrow 0\), we have the result.

Step 2.c. Take \(Z \sim \mu\) and \(Z' \sim \mu_{\infty}\), and call \(\psi^Z\) and \(\psi^{Z'}\) the solutions to equation

with initial conditions \(\psi (0) = Z\) and \(\psi (0) = Z'\), respectively. We show that \(\mu \sim \mu_{\infty}\).

Consider the product measure \(\mu \otimes \mu_{\infty}\) between \(\mu\) and \(\mu_{\infty}\). Since we are working in separable spaces, any Borel measure is a Radon measure (see, e.g. Theorem 9 in Chapter 2, Section 3 in [Sch73]), and therefore we have that, for every \(\varepsilon > 0\), there exists \(r > 0\) such that

Since

then we have, recalling \(\int \lambda G_{\lambda} \mathrm{d} \mu_{\infty} = \int F \mathrm{d} \mu_{\infty}\),

The right-hand side of the previous inequality converges to zero as \(\lambda \rightarrow + \infty\) by property (2.10), and thus \(\int F \mathrm{d} \mu_{\infty} = \int F \mathrm{d} \mu\) since, by equation (2.7) and being \(\mu\) a solution to Problem B, we have \(\lambda \int G_{\lambda} \mathrm{d} \mu = \int F \mathrm{d} \mu\). This yields \(\mu \sim \mu_{\infty}\) and uniqueness of the measure.\(\Box\)

2.2A priori deductions for exponential interaction

Let \(\alpha, m \in \mathbb{R}_+\), and \(\gamma := \alpha^2 / (4 \pi)\).

Let us introduce the functional \(\mathcal{G}_{\varepsilon} : B_X \times B_Y \rightarrow \mathcal{S}' (\mathbb{R}^2)\)

where the term \(: e^{\alpha (g_{\varepsilon} \ast X)} :\) is defined as

with the constant \(c_{\varepsilon}\) introduced in equation (1.4). Here, \(f_{\varepsilon}\) is a smooth cut-off function with compact support and with derivatives uniformly bounded in \(\varepsilon\) such that \(f_{\varepsilon} \rightarrow 1\) uniformly with respect to a polynomially weighted norm, and we let \(g_{\varepsilon} = \varepsilon^{- 2} g (\varepsilon^{- 1} \cdot)\), where \(g\) is a positive, smooth, compactly supported function on \(\mathbb{R}^2\) with Lebesgue integral equal to \(1\), and such that there exists its convolutional square root \(\tilde{g}_{\varepsilon}\), i.e. \(g_{\varepsilon} = \tilde{g}_{\varepsilon} \ast \tilde{g}_{\varepsilon}\), which is also positive, smooth, and compactly supported function with \(\int \tilde{g}_{\varepsilon} (x) \mathrm{d} x = 1\). We also assume that \(g_{\varepsilon} (x) = g_{\varepsilon} (- x)\), for all \(x \in \mathbb{R}^2\). Notice that this last property implies that \(\hat{g}_{\varepsilon}\) takes real values.

Consider an operator \(\mathcal{L}_{\varepsilon}\) of the form

which is well-defined for \(\Phi : B_X \times B_Y \rightarrow \mathbb{R}\) in a suitable class \(\mathcal{F}\) of regular functions to be specified below (see Definition 3.1). For the moment, we only require the set \(\operatorname{Cyl}_{B_X \times B_Y}\) of smooth cylindrical functions to be contained in the family \(\mathcal{F}\).

Remark 2.7. Our choice of \(B_X\) and \(B_Y\) implies that the space \(B_X\) contain the Gaussian free field with mass \(m\) and that

where all the parameters are as in Definition 2.6 (see [ADG21, HKK21, HKK22] and Appendix B). For simplicity, we adopt the notation

and we omit the parameters whenever \(\tilde{s} = r\) and \(\tilde{\ell} = \ell'\), i.e. \(B_{\text{exp}} = B^{\tilde{s}, \tilde{\ell}}_{\text{exp}}\).

Let us introduce also the notation

where the measurable function \(: \exp (\alpha X) :\) is defined on a subset of \(B_X\) of full measure (with respect to the free field measure with mass \(m\)) as the limit of \(: \exp (\alpha g_{\varepsilon} \ast X) :\) as \(\varepsilon \rightarrow 0\) (see Proposition B.1 for details). The operator \(\mathcal{L}_{\varepsilon}\) defined in equation (2.14) is an approximation for the operator

for \(\Phi \in \operatorname{Cyl}_{B_X \times B_Y}\).

Proof. We show the proof only for the case \(\Psi = 1\), the general case can be deduced via Lebesgue's dominated convergence theorem. Let us prove that \(\mathcal{L} \Phi \in L^1 (\mu)\), for \(\Phi \in \operatorname{Cyl}_{B_X \times B_Y}\). By definition, we have that \(\operatorname{tr}_{L^2 (\mathbb{R}^2)} (\nabla_X^2 \Phi)\) is bounded, and therefore in \(L^1 (\mu)\). Moreover,

and since \(\| X \|_{B_X} \in L^1 (\mu)\), \(P^X_{\ast} \mu\) being the Gaussian free field, and \(\| \nabla_X \Phi \|_{B_{1, 1, - \ell}^{2 - \delta}} \in L^{\infty} (B_X \times B_Y, \mathbb{R})\), then also the term \(\langle (- \Delta + m^2) X, \nabla_X \Phi \rangle\) is in \(L^1 (\mu)\). Now, the term \(\langle (- \Delta + m^2) Y, \nabla_Y \Phi \rangle\) can be handled similarly by using the hypothesis \(\| Y \|_{B_Y} \in L^1 (\mu)\).

We are left to consider the term \(\langle \mathcal{G}, \nabla_Y \Phi \rangle = \langle \alpha : e^{\alpha X} : e^{\alpha Y}, \nabla_Y \Phi \rangle\). First of all, we note that the product \(: \exp (\alpha X) : \exp (\alpha Y)\) is well-defined. Indeed, since \(P^X_{\ast} \mu\) is the Gaussian free field we can exploit Proposition B.1 to get \(: \exp (\alpha X) : \in B_{\exp}^{r, \ell'}\) (cf. Remark 2.7). Furthermore, since \(Y \leqslant 0\) and \(Y \in B^{s - \delta}_{p, p, \ell} (\mathbb{R}^2)\), by composition of a smooth bounded function with a Besov function we have \(\exp (\alpha Y) \in B^{s - \delta}_{p, p, \ell} (\mathbb{R}^2)\). Thus, by Theorem A.3, provided

we have that \(: \exp (\alpha X) : \exp (\alpha Y) \in B_{\exp}^{r', \ell + \ell'}\), and the product is well-defined and continuous. On the other hand, since we have also \(\exp (\alpha Y) \in L^{\infty} (\mathbb{R}^2)\), then Proposition A.4 implies that \(: \exp (\alpha X) : \exp (\alpha Y) \in B_{\exp}^{r, \ell}\) and

Thus,

By hypothesis we have \(\| \nabla_Y \Phi \|_{(B_{\exp}^{r, \ell})^{\ast}} \in L^{\infty} (B_X \times B_Y, \mathbb{R})\) and, since \(r \alpha^2 < 8 \pi\) by our assumptions in Definition 2.6, we also have that \(\| : \exp (\alpha X) : \|_{B_{\exp}^{r, \ell}} \in L^r (\mu)\). Hence,

We are left to show that equation (2.16) holds. This is equivalent to showing

By the same reasoning as in the previous part of the proof and by Proposition B.1, we have that \(\langle \mathcal{G}_{\varepsilon} -\mathcal{G}, \nabla_Y \Phi \rangle \in L^r (\mu)\), uniformly with respect to \(0 < \varepsilon < 1\). In order to show that the previous limit holds, it is sufficient to prove that

in probability as \(\varepsilon \rightarrow 0\). On the other hand, we know that

uniformly with respect to \(\varepsilon\), and by Proposition B.1 we have that \(\mathcal{G}_{\varepsilon} \rightarrow \mathcal{G}\), weakly in \(\mathcal{S}' (\mathbb{R}^2)\). Therefore, we have that \(\mathcal{G}_{\varepsilon} \rightarrow \mathcal{G}\) weakly in \(B_{\exp}^{r, \ell'}\), and in probability with respect to \(\mu\). By definition of weak convergence and since \(\nabla_Y \Phi \in (B_{\exp}^{r, \ell'})^{\ast}\) by hypothesis (see Definition 3.1), we have that \(\langle \mathcal{G}_{\varepsilon} -\mathcal{G}, \nabla_Y \Phi \rangle \rightarrow 0\), in probability. Finally, by property iv. in Definition 3.1 and the fact that

uniformly in \(\varepsilon > 0\) by Proposition B.1, we have that \(\langle \mathcal{G}_{\varepsilon} -\mathcal{G}, \nabla_Y \Phi \rangle \in L^p (\mu)\) uniformly in \(\varepsilon > 0\), for some \(p > 1\), and thus the result follows from Lebesgue's dominated convergence theorem.\(\Box\)

Remark 2.9. By Theorem 2.8 it is possible to take the limit \(\varepsilon \rightarrow 0\) inside the integration with respect to \(\mu\) in equations (2.4)–(2.3) and (2.5)–(2.6) so that the mentioned equations are equivalent to the one where the limit disappears and the operator \(\mathcal{L}_{\varepsilon}\) is replaced by \(\mathcal{L}\). Another consequence of the previous theorem is that, if we consider \(\tilde{\mathcal{L}}_{\varepsilon}\) to be another approximation of \(\mathcal{L}\) such that

then Problems A” and B with the operator \(\tilde{\mathcal{L}}_{\varepsilon}\) are equivalent to Problems A” and B with the operator \(\mathcal{L}_{\varepsilon}\). This means that the formulations given in Problems A” and B do not depend on the precise form of the approximating operator \(\mathcal{L}_{\varepsilon}\) but only on its limit \(\mathcal{L}\).

We now want to prove a result that justifies our restriction in taking \(B_Y = B_Y^{\leqslant}\) in the results above and in the rest of the paper. Indeed, if we focus on Problem B, the solutions are always supported on the space of negative functions on the \(Y\) component. In the light of point iii. in Theorem 2.5, the next result implies that any solution \(\mu\) to Problem B belongs to the spaces \(\mathcal{M}_{B_X \times B_Y}\).

Proof. Suppose that \(Y \in B_Y = B^{s - \delta}_{p, p, \ell} (\mathbb{R}^2)\) (see Definition 2.6). Define, for \(r, k > 0\),

and

We consider, for \(\eta > 0\), the convolution \(g_{\eta} \ast Y\) which, by definition of \(B_Y\), belongs to \(B^{s'}_{2, 2, \ell'}\) for any \(\ell', s' \geqslant 1\) such that \(p \ell' > 2\). Take

so that, by integration by parts,

Consider the term \(\langle f (g_{\eta} \ast Y), \nabla \rho^k_r (g_{\eta} \ast \nabla Y) \rangle\), and multiply and divide by \(\rho^k_r\) to get

By Young inequality, we have

where we also used that the term \(\langle f (g_{\eta} \ast Y), \rho^k_r (g_{\eta} \ast \mathcal{G}_{\varepsilon} (X, Y)) \rangle\) is positive. We want to show that \(\nabla \rho^k_r / \rho^k_r\) is bounded. Taking \(y = \sqrt{k} x\), we have, for \(k > 0\) small enough,

which gives, if \(C_k / 2 - m^2 < - \zeta\),

since \(f (g_{\eta} \ast Y) = g_{\eta} \ast Y\) on \(\operatorname{supp} (f)\).

Since the right-hand side does not depend on \(\varepsilon\), and the left-hand side converges to \(0\) when integrated with respect to \(\mu\), we have

and since the terms \(2^{- 1} \langle \rho^k_r, | g_{\eta} \ast \nabla Y |^2 \mathbb{I}_{\{ g_{\eta} \ast Y \geqslant 0 \}} \rangle\), \(\rho^k_r\), \(| g_{\eta} \ast \nabla Y |^2\), and \(\zeta (1 + \langle I (g_{\eta} \ast Y), \rho_{\ell'}^k \rangle^2)^{- 1}\) are all positive, we have

By Fatou's Lemma we can take the limit as \(\eta \rightarrow 0\) and get that

Remark 2.11. A consequence of Theorem 2.10 is that for the case of exponential interaction, a solution to Problem B with \(B_Y = B^{s - \delta}_{p, p, \ell} (\mathbb{R}^2)\) is necessarily a solution to Problem A” with \(B_Y = B_Y^{\leqslant}\).

2.3A description of the exponential measure

What has been described in the previous section has the main aim of giving a characterization to a unique measure which we call the exponential QFT. In this section, we want to connect our discussion of the IbP formula with the standard approach to construct EQFT via Gibbsian modifications of the GFF. Assume that \(\mu^{\operatorname{free}}_M\) is the measure of the free field with mass \(m\) on the torus \(\mathbb{T}^2_M\) of size \(M\). We define the interaction, for \(\varepsilon > 0\),

with \(f\), \(g\), \(c_{\varepsilon}\) and \(\alpha\) are as in Section 2.2. We also consider

Such a measure \(\nu^{\exp, \alpha}_m\) was first built by Albeverio and Høegh-Krohn in [AH74] (see also [FP77]) using techniques from constructive QFT. More recently, this model was studied in the context of stochastic quantization on the torus or on a compact manifold (see e.g. [AKMR21, BG21a, Gar20, HKK21, HKK22, ORTW20, ORW21]), and on \(\mathbb{R}^2\) (see [ADG21]). See also [BD21] for the related \(\cosh (\Phi)_2\)-model.

A consequence of the results we presented in the previous sections of the paper (see Theorem 1.2, Theorem 1.3) is the following differential characterization of the measure related to the exponential interaction.

3Uniqueness of solution

3.1Proof of uniqueness of solutions to Problem A”

In this section, we discuss the uniqueness of solutions to Problems A, A”, and B in the case where \(B_X\), \(B_Y\), \(\mathcal{M}=\mathcal{M}_{B_Y}\), and \(E = B_X + B_Y\) are as in Section 2.2 and \(B_{\varepsilon}\) is the drift of the exponential interaction (see equation (1.3)). The method that we adopt to prove uniqueness is the study of the resolvent equation for the operator \(\mathcal{L}_{\varepsilon}\) defined in equation (2.14), namely

We are interested in classical solutions to the previous equation, namely solutions \(G_{\varepsilon}^{\lambda}\) that are at least in \(C^2 (B_X \times B_Y)\) and to which the operator \(\mathcal{L}_{\varepsilon}\) can be applied.

Let us first introduce here a class of functions where the solutions of the resolvent equation belongs.

Remark 3.2. Under condition iii. of the previous definition, since the immersion \(H^{- \kappa}_{\ell} (\mathbb{R}^2) \hookrightarrow L^2 (\mathbb{R}^2)\) is an Hilbert-Schmidt operator, then \(\nabla_X^2 \Phi, \rho_{- \ell} \nabla_X^2 \Phi \in L_{\operatorname{loc}}^{\infty} (\hat{B}_X \times B_Y, \operatorname{TC} (L^2))\), where \(\operatorname{TC} (L^2)\) is the space of trace-class operators on \(L^2 (\mathbb{R}^2)\).

Remark 3.3. The classes \(\mathcal{F}\) and \(\mathfrak{F}\) are chosen in such a way that they satisfy the following two important properties: (i) \(\mathcal{F}\) and \(\mathfrak{F}\) contain \(\operatorname{Cyl}_{B_X \times B_Y}\) and \(\operatorname{Cyl}_E\), respectively, and (ii) if \(B_Y = B_Y^{\leqslant}\) then, for any Radon measure \(\mu \in \mathcal{M}_{B_X \times B_Y}\), we have \(\sup_{\varepsilon > 0} \int | \mathcal{L}_{\varepsilon} \Phi |^{\sigma} \mathrm{d} \mu < + \infty\), for any \(\Phi \in \mathcal{F}\) and \(\sigma \geqslant 1\).

The classes \(\mathcal{F}\) and \(\mathfrak{F}\) satisfy the following important lemma.

Proof. See Appendix D.\(\Box\)

Let \(\tilde{\gamma}_{\max}\) be the parameter defined in Remark 1.1. In the light of Proposition 3.5, the proof of which we postpone to Section 3.2, we can then proceed with the proof of uniqueness of solutions to Problem A”.

Proof. Let \(\mu_1\) and \(\mu_2\) be two solutions of Problem A”. We want to show that, for any \(F \in \operatorname{Cyl}_{B_X \times B_Y}\) with compact support in Fourier variables, \(\int F \mathrm{d} \mu_1 = \int F \mathrm{d} \mu_2\). This implies \(\mu_1 = \mu_2\) since \(F \in \operatorname{Cyl}_{B_X \times B_Y}\) with compact support in Fourier variables separates points of the space of Radon measures on \(B_X \times B_Y\). Consider the solution \(G_{\varepsilon}^{\lambda}\) of the resolvent equation (3.1) given by Proposition 3.5 and recall that, by point ii. in Proposition 3.5. Since \(\mu_1\) and \(\mu_2\) are solutions to the FPK equations we have in particular that equation (2.5) holds. Therefore, integrating with respect to \(\mu_1\) (respectively, with respect to \(\mu_2\)) the resolvent equation equation (3.1) and subtracting the integral \(\int \mathcal{L}G^{\lambda}_{\varepsilon} \hspace{1em} \mathrm{d} \mu_1 = 0\) (respectively, \(\int \mathcal{L}G^{\lambda}_{\varepsilon} \hspace{1em} \mathrm{d} \mu_2 = 0\)) which holds by Lemma 3.4, we get the relation

where we omitted the dependences on \((X_0, Y_0)\) for the sake of brevity. Taking the difference of such a relation for \(j = 1, 2\) yields

Now, point ii. in Proposition 3.5 gives us, for any \(\varsigma > 0\),

On the other hand, point iii. in Proposition 3.5 implies that

By the proof of Theorem 2.8, Proposition B.1, and using the fact that \(P^X_{\ast} \mu_j = \mu^{\operatorname{free}}\), we get that

Thus, if we take the limit \(\varepsilon \rightarrow 0\) and then \(\lambda \rightarrow 0\), we get \(\left| \int F \mathrm{d} \mu_1 - \int F \mathrm{d} \mu_2 \right| \lesssim \varsigma\), and since \(\varsigma > 0\) is arbitrary, we get \(\mu_1 = \mu_2\).\(\Box\)

In the case of Problem B a better result, in the sense that \(\gamma_{\max} > \tilde{\gamma}_{\max}\) (cf. Remark 1.1), can be achieved exploiting the properties of the resolvent operator proved in Section 3.3.

Proof. The proof is the same as in the case of Problem A”, but substituting point iv. of Proposition 3.5, with Proposition 3.11 below.\(\Box\)

3.2Analysis of the resolvent equation

To solve the resolvent equation as needed in Proposition 3.5, we use a probabilistic representation and look for solutions in the form

where \(X^{\varepsilon}_t\) and \(Y^{\varepsilon}_t\) are solutions to the stochastic differential system

where \(\mathcal{G}_{\varepsilon}\) is defined in equation (2.12), \(\xi\) is a Gaussian space-time white noise, and with initial conditions \((X_0, Y_0) \in B_X \times B_Y\). This is a triangular system, where the first equation does not depend on the second variable \(Y\). The first equation is a linear SDE with additive noise with stationary solution given by the Ornstein-Uhlenbeck process, whose invariant measure is the Gaussian free field with mass \(m > 0\). To prove that \(G^{\lambda}_{\varepsilon}\) is a classical solution we need some properties on the flow induced by the SPDEs (3.3)–(3.4). We will only state such properties in the present section – postponing their proof to Appendix E – and then proceed by showing that Proposition 3.5 holds.

First, let us write down the equations for the derivatives of the flow. Let us denote by \(X\) and \(Y\) the solutions to equations (3.3)–(3.4), dropping the dependence on the parameter \(\varepsilon > 0\) for simplicity of notation. The derivatives of \(X\) solve

Whenever \(Y_0 \in B^{\leqslant}_Y\),

for \(h \in B_Y \cup B_{\exp}^{r, \ell}\), and

for \(h \in B_X\), and

for \(h, h' \in L^2 (\mathbb{R}^2)\). The derivatives \(\mathrm{D} \mathcal{G}_{\varepsilon, \bar{\varepsilon}}\) have the following expressions

and

Hereafter, we will use the notation \(\gamma = \alpha^2 / (4 \pi)\), and

where all the appearing parameters are defined as in Definition 2.6.

The remainder of the present subsection is devoted to the proof of Proposition 3.5.

Proof of Proposition 3.5. Let us start by proving that \(G^{\lambda}_{\varepsilon}\) is a classical solution of the resolvent equation. We exploit Itô's formula appearing in Theorem 4.32 in [DZ14]. Notice that it can be applied to \(G^{\lambda}_{\varepsilon}\) since, by point i. in Proposition 3.8, \(G_{\varepsilon}^{\lambda}\) is a \(C^2\)-function with trace-class second derivative (continuity of the second derivative can be deduced with similar techniques as the ones adopted in Proposition E.7 and point i. in Proposition 3.5).

We use the notation \(X_t^{X_0}\) to denote a process \(X\) at time \(t\), with starting point \(X_0\) at time \(t = 0\). Recalling the definition of \(G^{\lambda}_{\varepsilon}\) in equation (3.2), we have

where \((\mathcal{H}_t)_{t > 0}\) is the filtration generated by the white noise at time \(t\) and the initial conditions \(X_0\) and \(Y_0\). By Markovianity of the process, we get

Since from now on \(X\) and \(Y\) always start at \(X_0\) and \(Y_0\), respectively, when \(t = 0\), we drop the dependence on the initial conditions. We have, by Itô's formula,

On the other side of the equation, we have

Notice that

and therefore

Dividing by \(t\) and letting \(t \rightarrow 0\), we get

Taking expectation also on the first side, we have the desired relation.

Proof of point i. When \(F \in \operatorname{Cyl}_{B_X \times B_Y}^b\), the derivative of \(G^{\lambda}_{\varepsilon} (X_0, Y_0)\) with respect to \(Y_0\) \(\)is given by

Therefore, we have

Applying the estimate (3.12), we get

Therefore,

We have that

where \(R_{2, \varepsilon}\) is a suitable second degree polynomial. Taking the supremum over \(h \in \tilde{B}_Y\) with \(| h | \leqslant 1\), recalling that \(\tilde{B}_Y^{\ast} \subset B^{(2 - s) \wedge (\gamma (r - 1)) + \delta}_{l, l, - \ell} (\mathbb{R}^2)\), for some \(l \in (1, + \infty)\), and taking \(f_{G_{\varepsilon}^{\lambda}} (X_0)\) as in Definition 3.1 to be proportional to \(\| f_{\varepsilon} \exp (\alpha | g_{\varepsilon} \ast X_0 |) \|_{L^{\infty}}^2 + 1\), we get that \(G_{\varepsilon}^{\lambda}\) satisfies the first condition of Definition 3.1. For the remaining conditions of Definition 3.1, similar arguments with the application of inequalities (3.13) and (3.14) allow us to conclude that \(G_{\varepsilon}^{\lambda} \in \mathcal{F}\).

Since in our approximation we have that \(\mathcal{G}_{\varepsilon} (X, Y) = \alpha f_{\varepsilon} e^{\alpha (g_{\varepsilon} \ast (X + Y)) - \frac{\alpha^2}{2} c_{\varepsilon}}\), then the sum \(\varphi_t^{\varepsilon} = X^{\varepsilon}_t + Y^{\varepsilon}_t\) solves the SPDE

Therefore, if \(F \in \operatorname{Cyl}^b_{B_X \times B_Y}\) is of the form \(F = \bar{F} \circ P^{X + Y}\) for some \(\bar{F} \in \operatorname{Cyl}^b_E\), we have

Proof of point ii. In the following, for \(j = 1, 2\), we denote by \((X^{\varepsilon, j}_t, Y^{\varepsilon, j}_t)\) the solution to the system of equations (3.3)–(3.4) with initial conditions \((X_0, Y_0) \sim \mu_j\). Sometimes we also write \((X_0^j, Y_0^j)\) to indicate that \((X_0, Y_0) \sim \mu_j\), for \(j = 1, 2\). By definition of \(G_{\varepsilon}^{\lambda}\) (i.e. by equation (3.2)), we have

Notice that, for any \(\varsigma > 0\), there exist two compact sets \(K_1, K_2\), such that \(\mu_1 (K_1), \mu_2 (K_2) > 1 - \varsigma\). Therefore, the following inequality holds

Let us then focus on the case where \((X^1_0, Y^1_0) \in K_1\) and \((X^2_0, Y^2_0) \in K_2\). By Lagrange's theorem, we have

where, for \(\beta \in (0, 1)\),

with \(\mathrm{d} F (X, Y) = \sum_{j = 1}^N \partial_j \tilde{F} \langle u_j, X \rangle + \sum_{j = 1}^M \partial_j \tilde{F} \langle v_j, Y \rangle\), for \(F \in \operatorname{Cyl}^b_{B_X \times B_Y}\) having the following form \(F (X, Y) = \tilde{F} (\langle u_1, X \rangle, \ldots, \langle u_N, X \rangle, \langle v_1, Y \rangle, \ldots, \langle v_M, Y \rangle)\), with \(\tilde{F} \in C^2_b (\mathbb{R}^{N + M})\) and \(u_i \in \mathcal{S} (\mathbb{R}^2)\), \(i = 1, \ldots, N\), and \(v_i \in \mathcal{S} (\mathbb{R}^2)\), \(i = 1, \ldots, M\), and, for \(\beta \in (0, 1)\),

We want to get some bounds for \(\left| \int_0^1 \langle \mathrm{d} F_{\beta}, \mathfrak{D}_{\beta} \rangle \mathrm{d} \beta \right|\).

For the term \(| \langle \mathrm{d} F_{\beta}, \nabla_{X^1_0 - X^2_0} X^{\varepsilon}_t ((1 - \beta) X^1_0 + \beta X^2_0) \rangle |\) we have

For the term \(| \langle \mathrm{d} F_{\beta}, \nabla_{Y^1_0 - Y^2_0} Y^{\varepsilon}_t ((1 - \beta) Y^1_0 + \beta Y^2_0) \rangle |\), considering equation (3.8), and multiplying it by \(\tilde{g}_{\varepsilon} \ast \nabla_{Y_0} Y_t\) and exploiting the negativity of the non-linearity, we get the following a priori estimate for some constant \(k \in (0, m^2)\)

and therefore, by Gronwall lemma,

and the estimate is independent of \(\varepsilon\).

Consider now the operator \(f \mapsto \tilde{g}_{\varepsilon} \ast f\), and recall that in Fourier representation convolution corresponds to multiplication, i.e. the previous operator can be viewed as \(\hat{f} \mapsto \widehat{\tilde{g}}_{\varepsilon} \cdot \hat{f}\). When \(\hat{f}\) has compact support, such an operator is invertible if \(0 < \varepsilon \leqslant \varepsilon_0\) for some positive constant \(\varepsilon_0\) depending on the size of the support of \(\hat{f}\), with inverse given by

which is a well-defined operation because \(\widehat{\tilde{g}}_{\varepsilon} (k) = \widehat{\tilde{g}}_1 (\varepsilon k)\), \(\widehat{\tilde{g}}_1 (0) = 1\), and \(\widehat{\tilde{g}}_1\) is smooth since \(\tilde{g}_1\) is a Schwartz function. Using the previous notation, we have

where we used the fact that \(F\) has compact support in Fourier variables and so the norm \(\| \mathrm{d} F_{\beta} \circ \mathcal{Q}_{\varepsilon} \|\) is bounded for any \(0 < \varepsilon \leqslant \varepsilon_0\), and also that

Now, focus on the term \(| \langle \mathrm{d} F_{\beta}, \nabla_{X^1_0 - X^2_0} Y^{\varepsilon}_t ((1 - \beta) Y^1_0 + \beta Y^2_0) \rangle |\). By equation (3.9), we have

and by Hölder and Young inequalities, and re-absorbing the terms properly, we have that, for any \(\varsigma > 0\), the following inequality holds

Now, we can bound the norm \(\| \nabla_{X^1_0 - X^2_0} X^{\varepsilon}_t ((1 - \beta) X^1_0 + \beta X^2_0) \|^2_{B^s_{q, q}}\) by a constant since \(X^1_0\) and \(X^2_0\) belong to compact sets. Moreover, by Lemma A.7, we have

Applying Gronwall lemma yields the following inequality

Thus, we get, for some constant \(C_{\varsigma} > 0\) depending on \(\varsigma > 0\),

since the law of \(X^{\varepsilon}_{\tau} ((1 - \beta) X^1_0 + \beta X^2_0)\) is that of a Gaussian free field independent of \(\beta \in (0, 1)\) and of \(\varepsilon > 0\).

Putting everything together, we get

where we used that the law of \((1 - \beta) X^1_0 + \beta X^2_0\) does not depend on \(\beta \in (0, 1)\).

Therefore, we get

For some constant \(K > 0\), we have

and the last term converges to zero as \(\lambda \rightarrow 0\). Thus, we have

Proof of point iii. Let \(\mathcal{K} \in B^{q, \ell / 2}_{\exp}\) and note that \(\nabla_{Y_0} Y^{\varepsilon}_t [\mathcal{K}]\) solves equation (3.8). We write

Recall that in the proof of point i. in Proposition 3.5 we saw that

Now, recalling the representation (3.15) of \(\nabla \mathcal{Y}^{\varepsilon}_t\), we have

By Lemma 3.9 below, we have

Hölder inequality with respect to all measures, together with the fact that \(\lambda > 0\), yields

Here, since \(\lambda > 0\), the integral \(\int_0^{+ \infty} e^{- \lambda t} (1 + (1 + t)^{\sigma}) \mathrm{d} t\) is finite. Moreover, by Proposition B.3, we have that the term

is bounded with respect to \(\varepsilon > 0\). This concludes the proof of Proposition 3.5.\(\Box\)

We close the section with an auxiliary lemma used in the previous proof.

Proof. Let \(\gamma = \alpha^2 / (4 \pi) < \tilde{\gamma}_{\max}\). The proof is similar to the one of Theorem C.1. Indeed, by multiplying the equation for \(\nabla \mathcal{Y}^{\varepsilon}\) by \(g_{\varepsilon} \ast \nabla \mathcal{Y}^{\varepsilon}\), and integrating, we get

In the last line, we have

Now, whenever \(\gamma < \tilde{\gamma}_{\max}\), there exist \(p, q \geqslant 1\), \(0 \leqslant \theta \leqslant 1\), \(0 \leqslant \beta \leqslant 1\), \(\delta > 0\), such that the following system has a solution

The proof is complete.\(\Box\)

3.3Proof of uniqueness of solutions to Problem B

In the case of Problem B a better result can be obtained. It is useful to introduce a new approximating operator \(\mathcal{L}_{\varepsilon, \bar{\varepsilon}}\) for \(\mathcal{L}\) of the form

where

Here \(f_{\varepsilon}\) and \(g_{\varepsilon}\) are defined as in Section 2.2. Recall that by Remark 2.9, since \(\lim_{\bar{\varepsilon}, \varepsilon \rightarrow 0} \mathcal{L}_{\varepsilon, \bar{\varepsilon}} =\mathcal{L}\), Problem B with the operator \(\mathcal{L}_{\varepsilon, \bar{\varepsilon}}\) is equivalent to the one with operator \(\mathcal{L}_{\varepsilon}\) defined in equation (2.14). We can then consider the resolvent equation associated to \(\mathcal{L}_{\varepsilon, \bar{\varepsilon}}\), namely

where \(F \in \operatorname{Cyl}_{B_X \times B_Y}\) with compact support in Fourier variables. A solution to such equation is given by

where \(X_t, Y^{\varepsilon, \bar{\varepsilon}}_t\) solves the following system of equations

It is easy to show that all the results in Proposition 3.8 hold also for equations (3.17)–(3.18), adapting the form of the equations for the derivatives. This implies that points i., ii. stated in Proposition 3.5 hold true also for \(G^{\lambda}_{\varepsilon, \bar{\varepsilon}}\). As far as point iii in Proposition 3.5 is concerned, we can get a slightly better result in the present scenario. Recall that \(\gamma_{\max}\) is defined as in Remark 1.1.

Notice that the operator (3.16) and equations (3.17)–(3.18) can be defined also for the case \(\bar{\varepsilon} = 0\). Moreover, (a suitable adaptation of) Proposition 3.8 holds also for the case \(\bar{\varepsilon} = 0\), and point iii. of Proposition 3.5 holds also for \(G^{\lambda}_{\varepsilon, 0}\).

Remark 3.10. It is worth to note that the operator \(\mathcal{L}_{\varepsilon, \bar{\varepsilon}}\) cannot be use directly to solve Problem A”, since the solution to the resolvent equation depends in general on \((X, Y)\) and not only on \(X + Y\).

In order to prove the previous result, we need some technical lemmas. First, we deal with the convergence as \(\bar{\varepsilon} \rightarrow 0\). In particular, whenever an object has \(\bar{\varepsilon}\) as one of its parameters (e.g. \(\nabla_{Y_0} G^{\lambda}_{\varepsilon, \bar{\varepsilon}}\)), the same notation with \(\bar{\varepsilon} = 0\) indicates that it is the limiting object as \(\bar{\varepsilon} \rightarrow 0\) (e.g. \(\nabla_{Y_0} G^{\lambda}_{\varepsilon, 0} = \lim_{\bar{\varepsilon} \rightarrow 0} \nabla_{Y_0} G^{\lambda}_{\varepsilon, \bar{\varepsilon}}\)), whenever it exists.

Proof. Recall that \(\nabla_{Y_0} Y^{\varepsilon, \bar{\varepsilon}}_t (\mathcal{G}-\mathcal{G}_{\varepsilon, \bar{\varepsilon}})\) solves equation (3.8). This means that

solves the equation

By Corollary C.3, we get the estimate

Such a bound is uniform with respect to \(\bar{\varepsilon} > 0\) and therefore we get a converging subsequence whose limit \(\nabla \mathcal{Y}_t^{\varepsilon, 0}\) solves equation (3.19) with \(\bar{\varepsilon} = 0\). Since the term \(P_t (\mathcal{G}-\mathcal{G}_{\varepsilon, \bar{\varepsilon}})\) converges point-wise to \(P_t (\mathcal{G}-\mathcal{G}_{\varepsilon, 0})\), the result is proved.\(\Box\)

Let us prove that it makes sense to consider some parameters satisfying certain conditions that will be useful in upcoming results.

Proof. In order to prove the result, it is enough to show that there exists some solution with the previous properties to the system of equations

From such relations we get the equality

We get the maximum value \(\gamma_{\max}\) of \(\gamma\) for some \(r = \bar{r} \approx 2.52\). Since, with this choices of parameters, we have \(q \approx 1.21\), the result is proved.\(\Box\)

In the proof of Lemma 3.12, we introduced the object

satisfying equation (3.19) and admitting a limit \(\nabla \mathcal{Y}_t^{\varepsilon, 0}\) as \(\bar{\varepsilon} \rightarrow 0\).

Proof. Let \(\gamma = \alpha^2 / (4 \pi)\). With the same notation as in Lemma 3.12, we consider the limit \(\nabla \mathcal{Y}^{\varepsilon, 0}_t\) and show that it converges to zero. Indeed, it satisfies

with \(\nabla \mathcal{Y}^{\varepsilon, 0}_0 = 0\).

We now want to exploit a similar argument as the one used in Theorem C.2 to get some estimates concerning the solution to the previous equation. Let \(w : \mathbb{R} \rightarrow \mathbb{R}\) be an increasing smooth function with bounded derivatives and such that \(w (0) = 0\), \(w (x) \rightarrow \pm 1\) as \(x \rightarrow \pm \infty\), and define \(W (x) = \int_0^x w (y) \mathrm{d} y\). Then, multiplying the equation by \(\rho_{\ell} (\theta \cdot) w (\tilde{\mathcal{K}}_t^{\varepsilon, 0})\), where \(\theta, \ell > 0\), and integrating, we have

where \(0 < m_{\theta} \leqslant \sqrt{m^2 - \left| \frac{\nabla \rho_{\ell} (\theta z)}{\rho_{\ell} (\theta z)} \right|}\) for every \(z \in \mathbb{R}^2\), which holds for \(\theta > 0\) small enough.

If we choose the parameters as in Lemma 3.13, we have

Therefore, by integrating with respect to time equation (3.23) and recalling that \(\nabla \mathcal{Y}^{\varepsilon, 0}_0 = 0\), we get

where we used the fact that \(\frac{1}{r} + \frac{1}{q} < 1\) and \(- \gamma (r - 1) - \gamma (q - 1) - 2 \delta + \kappa > 1\). By Proposition A.4, we have that

Therefore, by Proposition A.6, we have

Multiply and divide then by \(\rho_{\ell'''} (s)\) and apply Hölder inequality with respect to \(s\) to get

Since the expression \(\left( \int_0^t s^{q \kappa / 2} \rho_{- \ell''' q} (s) \mathrm{d} s \right)^{1 / q}\) is bounded for \(t \rightarrow 0\) and grows polynomially in time as \(t \rightarrow + \infty\), we deduce that the result holds.\(\Box\)

Proof of Proposition 3.11. As in the proof of point i. of Proposition 3.5 we can get

Therefore, by Lemma 3.12 and Lebesgue's dominated convergence theorem, we have

Now, recalling the representation (3.22) of \(\nabla \mathcal{Y}^{\varepsilon, 0}_t\), we have

At this point, the proof follows the same steps as the one of point iv. in Proposition 3.5.\(\Box\)

4Existence of solutions via Lyapunov functions

In this section, we prove existence of solutions to Problem A” and B. Since existence for the Problem B implies existence for Problem A” (see point iii. in Theorem 2.5), we only prove the first. We actually prove a stronger statement than the one given in Problem B, namely that equation (2.5) holds for every \(\Phi \in \mathcal{F}\) and not only for \(\Phi \in \operatorname{Cyl}_{B_X \times B_Y}\). We exploit a strategy based on Lyapunov functions.

Let us introduce a finite-dimensional approximation to the operator \(\mathcal{L}\). Let \(M, N \in \mathbb{N}\), \(\varepsilon > 0\), and consider \(\mathbb{T}^2_M\) the two-dimensional torus of size \(M\) which we identify hereafter with the subset \((- M \pi, M \pi]^2 \subset \mathbb{R}^2\). Consider the Fejér operator \(Q_{N, M} : \mathcal{S}' (\mathbb{T}^2_M) \rightarrow C^{\infty} (\mathbb{T}^2_M)\) defined as, for \(F \in \mathcal{S}' (\mathbb{T}^2_M)\),

Let us stress that the operator \(Q_{N, M}\) is both a positive operator, i.e. \(\langle Q_{N, M} (F), F \rangle \geqslant 0\), and a positive preserving operator, namely, if \(F\) is a positive distribution, then \(Q_{N, M} (F)\) is a positive function. The latter property following from the positivity of the kernel \(\operatorname{Fej}_{N, M}\).

The new approximating operator \(\mathcal{L}_{N, M, \varepsilon}\) is then given by, for \(\Phi \in \mathcal{F}\), by

where \(\mathcal{G}_{N, M, \varepsilon}\) is the following approximation for the non-linearity in equation (3.4),

Here, \(g_{\varepsilon}\) is defined as in Section 2.2, \(P_{\mathbb{T}^2_M}\) is the natural projection of \(L^2 (\mathbb{R}^2)\) on the space \(L^2 (\mathbb{T}^2_M)\), and \(\operatorname{Per}_{\mathbb{T}^2_M}\) is the periodicization on the torus \(\mathbb{T}^2_M\). Moreover, the term \(: \exp (\alpha Q_{N, M} (g_{\varepsilon} \ast X)) :\) is defined as follows

where \(c_{N, M, \varepsilon} = \int_{\mathbb{T}^2_M} Q_{N, M} (g_{\varepsilon}) (z) (- \Delta + m^2)^{- 1} (Q_{N, M} g_{\varepsilon}) (z) \mathrm{d} z\), where the inverse \((- \Delta + m^2)^{- 1}\) is taken with periodic boundary conditions on \(\mathbb{T}^2_M\).

The system of equations for the flow is then given by

where \(Y^{N, M, \varepsilon}\) is negative, periodic, and it belongs to a subspace of \(\mathfrak{H}_{N, M} := \operatorname{span} \{ e^{i n \cdot / M} ; | n | \leqslant N \}\). As usual, we drop the dependence on the parameters \(N, M, \varepsilon\) when no ambiguity occurs.

4.1Lyapunov functions

We introduce some Lyapunov functions \(V_1\), \(V_2\), and \(V_3\) that will be crucial in the proof of the estimates for \(X\) and \(Y\). In particular, we take them such that \(V_1\) and \(V_2\) depend on both \(X\) and \(Y\), while \(V_3\) depends on \(X\) only.

Proof. The statement follows from the fact that \(\int \tilde{\mathcal{L}} V_1 (X, Y) \tilde{\mu} (\mathrm{d} X, \mathrm{d} Y) = 0\).\(\Box\)

We now work on the estimate for \(X\). We will have to choose the aforementioned Lyapunov functions, we start with \(V_1 : B_X \rightarrow \mathbb{R}\), that we take to be, for \(s' > 0\),

Note that, since \(X\) corresponds to the free field, the previous norm is finite. Notice also that the previous function is not cylindric. Moreover, if we consider the representation of Besov spaces given in Proposition A.9, then we have, for \(- p s' / 2 + 1 < 0\),

where \(k\) is to be determined, similarly as in the proof of Proposition 2.10.

Proof. We want to evaluate \(\mathcal{L}_{N, M, \varepsilon} V_1 (X)\). The gradient of \(V_1\) is given by

while its second derivative is

The first term to deal with is \(\langle (- \Delta + m^2) X, \nabla_X V_1 (X) \rangle\). We have

Rearranging the second term on the right-hand side and exploiting Young's inequality, we have, for all \(\sigma > 0\),

from which, choosing \(\sigma > 0\) small enough and \(k > 0\) large enough, and appropriately rearranging the terms, we get

We also have

The second term to deal with is the trace of the second derivative, i.e. \(\operatorname{tr}_{L^2 (\mathbb{T}^2_M)} (\nabla^2 V_1 (X))\). By the results presented in Section XI.V of [RS79], it suffices to consider the second derivative with \(h' = h\), i.e. \(\nabla^2 V_1 (X) (h, h)\), and integrate with respect to \(h\). Then, exploiting Young inequality for products and using the fact that \(p \ell > 2\), we have for any \(\sigma > 0\)

We get, if \(\mathcal{P}^M_t\) is the heat kernel on the torus \(\mathbb{T}^2_M\), that, for any \(\sigma > 0\),

where we used Young inequality as above.

Choosing \(\sigma > 0\) small enough, we get

which gives the result.\(\Box\)

Remark 4.3. The result in Lemma 4.1 with \(B_2 = B_X\), no dependence on \(B_1\) whatsoever, and with Lyapunov functions chosen as

for some constant \(C > 0\), gives the estimate

Notice further that the bound (4.6) can be chosen to be uniform with respect to the size \(M\) of the torus \(\mathbb{T}^2_M\) since the integrals \(\int_{\mathbb{T}^2_M} | \hat{\varphi}_0 (z - y) |^2 \mathrm{d} y\) and \(\int_{\mathbb{T}^2_M} | | \mathcal{P}^M_t (z - y) | (z - y) |^2 \mathrm{d} y\) are uniformly bounded in \(M\). It is worth to note also that by the Besov embedding (Proposition A.2) the space \(B^{- s' + 2 / p}_{p, p, \ell} (\mathbb{R}^2)\) is embedded in \(B_X = C^{- \delta}_{\ell} (\mathbb{R}^2)\).

Let us choose the Lyapunov functions in order to get an estimate for \(Y\). We may need \(X \mapsto V_1 (X, \cdot)\) to be \(C^1\) and \(Y \mapsto V_1 (\cdot, Y)\) to be \(C^2\).

Let \(s > 0\). Referring to the representation of Besov spaces given in Proposition A.9 and Remark A.10, take \((k - s / 2) p > 1\), and consider

Proof. In the following proof, we neglect the term \(V_{1, 1} (Y)\) appearing in (4.7), which can be dealt with in a similar way as in the proof of Proposition 4.2. The gradient of \(V_1\) is given by

Therefore, taking \(h = - (- \Delta + m^2) Y -\mathcal{G}_{N, M, \varepsilon} (X, Y)\), we have

Let us focus on \(I_1\). Integrating by parts, we get

Consider now \(I_2\). Exploiting Hölder inequality with \(q_1, p_1\) and introducing \(s' > 0\) yields

If we apply Young inequality, we then have, for any \(\sigma > 0\),

We want to apply Besov embedding (see Proposition A.2) in order to reabsorb the term \(I_{2, 1}\) in \(I_1\): considering the parameters of the involved norms, we get the condition

so that the reabsorbing procedure follows by a suitable choice of \(p_2\).

On the other hand, we have by Proposition A.4

All in all, we have the inequality

provided condition (4.10) holds true and \(p_2\) is chosen in an appropriate way.

Now, in order to obtain (4.9), we consider \(q_1 = r\) and \(s' > \gamma (r - 1)\), and have to check the condition

which is satisfied with the choice of parameters given in (4.8).\(\Box\)

Remark 4.5. If \(Y \leqslant 0\), we have the bound

on the right-hand side of inequality (4.9). In that case, choosing the Lyapunov functions as follows

we get that the conditions in Lemma 4.1 are satisfied thanks to Theorem 4.4. Therefore, we have

4.2Measure of the approximating problem

Equations (4.3) and (4.4) induce a Markov process on \(B_X \times B_Y^{\leqslant}\). This is due to the fact that, if we start from a negative initial condition for \(Y\), then \(Y\) remains negative throughout all the evolution.

Proof. The proof is based on the fact that we can apply Itô formula to the functions in \(\mathcal{F}\). Exploiting Itô formula, the proof follows the argument of the proof of point ii. in Proposition 3.5.\(\Box\)

By linearity, the invariant measure for equation (4.3) is given by the free field measure with mass \(m\) on the torus \(\mathbb{T}^2_M\), that we can identify as usual with the periodic free field of mass \(m\) on the whole space \(\mathbb{R}^2\).

We are interested in studying (infinitesimally) invariant measures for the aforementioned pair of equations, i.e. equations (4.3) and (4.4). Indeed, by Proposition 4.6, an invariant measure for those two equations is a solution to a FPK equation associated with \(\mathcal{L}_{N, M, \varepsilon}\) in the sense of Problem B. Let us start off by giving an argument for the existence of an infinitesimally invariant measure (for its definition and the relation with invariant measures see Chapter 5 in [BKRS15]) by means of the Lyapunov functions introduced in Section 4.1.

Proof. The proof follows the same argument as Lemma 3.3 in [ADRU22]. We start equations (4.3) and (4.4) from deterministic initial conditions \((x_0, y_0) \in B_X \times B_Y^{\leqslant}\), and let \(\mu_{(x_0, y_0), t}\) denote the measure of the solution of the equations starting at \((x_0, y_0)\). Let

If such a measure is tight with respect to \(T\), and \(\tilde{\mu}\) is its weak limit as \(T \rightarrow + \infty\), then \(\tilde{\mu}\) is an infinitesimal invariant measure of the equation. We consider the following Lyapunov functions, consisting of the sum of the one considered in Section 4.1 where we dealt with \(X\) and \(Y\) separately (see Remarks 4.3 and 4.5), namely

Then, by Proposition 4.2 and Theorem 4.4, we have

Recall that \(X_{\tau} = e^{- (- \Delta + m^2) \tau} x_0 + \int_0^{\tau} e^{- (- \Delta + m^2) (\tau - s)} \xi_s \mathrm{d} s\), then \(\mathbb{E} \left[ \|\mathcal{G}_{N, M, \varepsilon} \left( {X_{\tau}} , 0 \right) \|^{(p r - r + 1) / (p r^2)}_{B_{\exp}^{r, \ell}} \right]\) converges exponentially to a constant, and hence

Therefore, we have, for any \(t > 0\),

which yields

Taking the supremum over \(t \in [0, + \infty)\), we have

On the other hand, for any \(t > 0\),

Since \(V_2 (X, Y)\) has compact sub-levels on \(B_X \times B_Y^{\leqslant}\), tightness of \(\tilde{\mu}_t\) is implied.\(\Box\)

We can actually prove a stronger result on the form of one invariant measure for equations (4.3) and (4.4). Indeed, if we start from \(x_0 = 0\) and \(y_0 = 0\) in the proof of Proposition 4.7, then we have that the invariant measure that has been built there must be the measure of the process solving the equation

It is possible, in fact, to prove that the previous integral equations admit a unique solution.

Proof. Existence and uniqueness for equation (4.14) can be proved in the same way as in the proof of Theorem E.1.

In order to prove the second part of the result, we exploit the fact that

Indeed, if \(X_t^{0, - T}\) and \(Y_t^{0, - T}\) are the processes solution to equations (4.3) and (4.4) starting at time \(- T\) with initial condition zero, then we have that

and that \((X_t^{0, - T}, Y_t^{0, - T}) \rightarrow (X_t, Y_t)\) as \(T \rightarrow + \infty\), almost surely. Therefore, by the proof of Proposition 4.7, the law of \((X_t, Y_t)\) is an invariant measure for equations (4.3) and (4.4), and thus a solution to the FPK equation for \(\mathcal{L}_{N, M, \varepsilon}\).\(\Box\)

Hereafter, we denote by \(\mu_{N, M, \varepsilon}\) the law at fixed time \(t\) of the process \((X_t, Y_t)\) solution to equation (4.14), which is a solution to the FPK equation for \(\mathcal{L}_{N, M, \varepsilon}\). Thanks to the representation given by Proposition 4.8, we are able to establish some more precise properties of the measure \(\mu_{N, M, \varepsilon}\).

Proof. Property i. follows from the fact that the solution \(Y_t\) to the second equation in (4.14) is supported on the image of the projection of the operator \(Q_{N, M}\), which is exactly \(\mathfrak{H}_{N, M}\).

Property ii. is due to the fact that \(\mu_{N, M, \varepsilon}\) solves the FPK equation for \(\mathcal{L}_{N, M, \varepsilon}\). Indeed, the system (4.14) can be split in an infinite-dimensional linear equation and an independent finite-dimensional non-linear equation. The statement then follows from the results in Section 2.3 in [Da 04] for the infinite-dimensional part and Theorem 5.2.2 in [BKRS15] for the non-linear finite-dimensional part.

As far as point iii.\(\) is concerned, letting \(Z_t = X_t + Y_t\) and calling \(Z_t^1\) the projection of \(Z_t\) on \(\mathfrak{H}_{N, M}\) and \(Z_t^2\) the projection on \(\mathfrak{H}_{N, M}^{\bot}\), we have that \(Z_t^1\) and \(Z_t^2\) solve two independent equations: the equation for \(Z_t^1\) is a non-linear finite-dimensional equation, since \(\mathfrak{H}_{N, M}\) is a finite-dimensional Hilbert space a linear equation, while the equation for \(Z_t^2\) is a linear equation in infinite dimensions. More precisely, we have

where \(\delta\) is the functional derivative of

with \(c_{N, M, \varepsilon}\) is the constant appearing in the definition (4.2) of the Wick product appearing in the functional \(\mathcal{G}_{N, M, \varepsilon}\), and \(\)\(\xi^1_t\) and \(\xi^2_t\) are the projection of \(\xi\) on the spaces \(\mathfrak{H}_{N, M}\) and \(\mathfrak{H}_{N, M}^{\bot}\), respectively.

Since the equation for \(Z_t^1\) is a finite-dimensional equation with drift given by the gradient of a function and the equation for \(Z_t^2\) is linear with self-adjoint drift, then the unique invariant measure of the process \((Z^1, Z^2)\) satisfies property iii., and hence also their sum \(Z^1 + Z^2 = X + Y\) does. For further details on the relation between symmetric processes and the integration by parts formula see e.g. [AR91, Bog10, Föl88].\(\Box\)

Remark 4.10. From the proof of the previous result it is evident that the system consists of two independent equations: an infinite-dimensional linear one and a finite-dimensional non-linear one. This means that \(\mu_{M, N, \varepsilon}\) is the unique invariant measure of the system.

4.3Tightness of the measure

We prove tightness of the measures \(\mu_{N, M, \varepsilon}\).

Proof. Let \(V_1\), \(V_2\), and \(V_3\) be the Lyapunov functions defined in equations (4.11), (4.12), and (4.13), respectively. Since the measures \(\mu_{N, M, \varepsilon}\) are solutions to the FPK equation associated with \(\mathcal{L}_{N, M, \varepsilon}\) in the sense of Problem B and \(V_1 \in \mathcal{F}\), then

Moreover, we have by Theorem 4.2 together with Remark 4.3 and by Theorem 4.4 together with Remark 4.5, that \(V_1\) is a Lyapunov function for \((X, Y)\), where \(X\) and \(Y\) solves equation (4.3) and (4.4), respectively. Thus, by Lemma 4.1, we have

Since \(\sup_{M \in \mathbb{N}} \int V_3 (X) \mu^{\operatorname{free}}_M (\mathrm{d} X) < + \infty\), and since \(V_2\) has compact sub-levels, the thesis follows.\(\Box\)

We get the following consequence.

Proof. The result follows from Theorem 4.11 and a diagonalization argument.\(\Box\)

4.4Limit of the operator

We want now to prove that any limit measure \(\mu\) built in Corollary 4.12 solves the FPK equation in the sense of Problem B. We first prove that any measure appearing in Corollary 4.12 solves the FPK equation associated with the corresponding operator. For simplicity, let us drop the dependence on the parameters \(k\), \(l\), and \(j\) introduced in Corollary 4.12.

The aim of the section is to prove the following result.

In order to prove the limit (4.15), we proceed by first taking \(N \rightarrow + \infty\), then \(M \rightarrow + \infty\), and eventually \(\varepsilon \rightarrow 0\), showing the convergence step by step.

4.4.1Limit as \(N \rightarrow + \infty\)

As mentioned above, we start off by taking \(N \rightarrow + \infty\), namely we want to show that

The term in the operator \(\mathcal{L}_{M, N, \varepsilon}\) involving \(X\) only is independent of \(N\), in particular we do not need to consider the trace. Instead, we need to work on the terms involving only the derivatives with respect to \(Y\). Since, by Corollary 4.12, \(\mu_{N, M, \varepsilon}\) is tight and converges weakly up to subsequences to some limit \(\mu_{M, \varepsilon}\), we have to show that

Let us compute the integrand and rearrange its terms. We have

with

and

Let us deal with the term \(\mathbb{I}\) (4.16). We need, for \(p > 1\), the bound

Indeed, if estimate (4.19) holds, then, by the regularization property of \(g_{\varepsilon}\) – namely the continuity for any \(s > 0\) of the operator \(g_{\varepsilon} \ast (\cdot) : L^{\infty} (\mathbb{T}^2_M) \rightarrow B^s_{\infty, \infty} (\mathbb{T}^2_M)\) – and exploiting the fact that the norm \(\| Q_{N, M} - I \|_{L (B^s_{\infty, \infty}, B^{s - \delta}_{\infty, \infty})}\) converges to zero as \(N \rightarrow + \infty\) for any \(s > 0\) and \(\delta > 0\), we get the convergence of the term \(\mathbb{I}\) to \(0\) as \(N \rightarrow + \infty\). Let us prove the bound (4.19). The exponential involving \(Y\) disappears, since \(Y \leqslant 0\). Moreover, since \(\varepsilon > 0\), we have that the Wick exponential can be written as the exponential divided by some positive constant \(C_{N, M, \varepsilon}\), i.e.

which converges to some finite number as \(N \rightarrow + \infty\). Then, we have the inequality

and hence, by positivity of \(Q_{N, M}\), we obtain

which is in \(L^p (\mu_M^{\operatorname{free}})\) by Fernique's theorem (see Theorem 2.8.5 in [Bog98]).

We deal with the term \(\mathbb{I}\mathbb{I}\) given by (4.17). We have

and, by stochastic estimates (see Proposition B.2), we have convergence of the mean of the term \(\mathbb{I}\mathbb{I}\) to zero.

For the term \(\mathbb{I}\mathbb{I}\mathbb{I}\) given by (4.18) we exploit the non-positivity of \(Y\) and replace the exponential by a bounded smooth function with bounded derivatives. Then we have that

provided

On the other hand, we have the bound

But \(\| Q_{N, M} - I \|_{L (C^{\delta}, L^{\infty})} \rightarrow 0\) depending only on \(M\), while \(\| g_{\varepsilon} \ast Y \|_{C^{\delta}} \in L^p\) uniformly since by the proof of Theorem 4.11 we have that the integral \(\int V_2 (X, Y) \mu_{N, M, \varepsilon} (\mathrm{d} X, \mathrm{d} Y)\) is bounded uniformly with respect to \(M\).

4.4.2Limit as \(M \rightarrow + \infty\)

We now have to take \(M \rightarrow + \infty\), namely we want to show that

Notice that the only \(M\)-dependent term is now the trace-term \(\operatorname{tr}_{L^2 (\mathbb{T}^2_M)} (P_{\mathbb{T}^2_M} \nabla_X^2 \Phi P_{\mathbb{T}^2_M})\). By the same considerations as above, we have to show

Since \(\Phi \in \mathcal{F}\) (see Definition 3.1 and Remark 3.2), we have

where \(\int f_{\Phi} \mathrm{d} \mu^{\operatorname{free}} < + \infty\). First, note that

Therefore, we have the bound

Now, let \(h \in L^2 (\mathbb{R}^2)\), then we have

Take \(\ell > \ell' > 1\), then, considering the operator \(L^2\) norm, we have

Since \(\rho_{\ell} (z - y) = \rho_{\ell'} (z - y) \rho_{\ell - \ell'} (z - y) \leqslant \rho_{\ell'} (z - y) (1 + M)^{- (\ell - \ell')}\), we get

Now note

and \(\operatorname{Per}_{\mathbb{T}^2_M} (\rho_{\ell'}) (z) \leqslant C_{\ell'} := \sum_{y \in \mathbb{Z}^2} \rho_{\ell'} (y)\), to get

By taking the supremum over all \(h\) with \(\| h \|_{L^2 (\mathbb{R}^2)} \leqslant 1\), we then obtain

Letting \(M \rightarrow + \infty\), we get the limit (4.20).

4.4.3Limit as \(\varepsilon \rightarrow 0\)

Now, we are left to show the convergence

As above, we have to show that

Rewriting the integrand, we get

with

The stochastic estimates for \(: \exp (\alpha (g_{\varepsilon} \ast X)) :\) are done in Proposition B.1. We now deal with the term \(\mathbb{I}'\) (4.22). We have the inequality

Taking the expectation and exploiting the negativity of \(Y\), we have

and this last term converges to zero as \(\varepsilon \rightarrow 0\). The convergence to zero of the term \(\mathbb{I}\mathbb{I}'\)\(\) (4.23) follows from Proposition B.1. Finally for the term \(\mathbb{I}\mathbb{I}\mathbb{I}'\)\(\) (4.24), we proceed as follows. We have the bound

Since \(\mu_{\varepsilon}\) is tight and \(\| : \exp (\alpha X) : \|_{B^s_{p, p, \ell}}\) is uniformly integrable with respect to the measure \(\mu_{\varepsilon}\), then, for any \(\tau > 0\), there exists a Borel subset \(\Omega_{\tau} \subset B_X \times B_Y\) such that

and \(R_{\Omega_{\tau}} = \sup_{\Omega_{\tau}} \| Y \|_{B^{- s + \delta}_{q, q, \ell}} < + \infty\). Therefore, we have

Thus,

which gives convergence to zero by arbitrary choice of \(\delta\). In order to show inequality (4.25), note that

where \(l \in (1, + \infty)\) and \(1 / l' + 1 / l = 1\) such that \(\delta > 2 - 2 / l'\).Now, if \(| s |\) is small enough such that \(B^{- s + \delta}_{q, q, \ell} \subset B_Y\), then we have the

which is converging to zero for every \(Y \in\)\(B^{- s + \delta}_{q, q, \ell} \subset B_Y\). We conclude the argument by Lebesgue dominated convergence theorem and with the reasonings similar as above.

Appendix ABesov spaces and heat semigroup

In this section, we collect some results about weighted Besov spaces. While we only focus on spaces defined on the whole space \(\mathbb{R}^n\), the results hold also for Besov spaces on the \(n\)-dimensional torus \(\mathbb{T}^n\).

Let us start by introducing Littlewood-Paley blocks. Let \(\chi\) and \(\varphi\) be smooth non-negative functions from \(\mathbb{R}^n\) into \(\mathbb{R}\) satisfying the following properties:

• \(\operatorname{supp} (\chi) \subset B_{4 / 3} (0)\) and \(\operatorname{supp} (\varphi) \subset B_{8 / 3} (0) \setminus B_{3 / 4} (0)\),

• \(\chi, \varphi \leqslant 1\) and \(\chi (y) + \sum_{j \geqslant 0} \varphi (2^{- j} y) = 1\), for any \(y \in \mathbb{R}^n\),

• \(\operatorname{supp} (\chi) \cap \operatorname{supp} (\varphi (2^{- j} \cdot)) = \varnothing\), for \(j \geqslant 1\),

• \(\operatorname{supp} (\varphi (2^{- j} \cdot)) \cap \operatorname{supp} (\varphi (2^{- i} \cdot)) = \varnothing\), if \(| i - j | > 1\),

where \(B_r (x)\) denotes the ball centered at \(x \in \mathbb{R}^n\) with radius \(r > 0\).

We introduce the following notations: \(\varphi_{- 1} = \chi\), \(\varphi_j (\cdot) = \varphi (2^{- j} \cdot)\), \(D_j = \hat{\varphi}_j\), and for any \(f \in \mathcal{S}' (\mathbb{R}^n)\) we put \(\Delta_j (f) = D_j \ast f\). Moreover, we write, for any \(\ell > 0\), \(\rho_{\ell} (y) = (1 + | y |^2)^{- \ell / 2}\), and let \(L^p_{\ell} (\mathbb{R}^n)\) be the \(L^p\)-space with respect to the norm

where \(p \in [1, + \infty]\).

In the case where \(p = q = + \infty\), the weighted Besov space \(B^s_{\infty, \infty, \ell} (\mathbb{R}^n)\) is denoted by \(C^s_{\ell} (\mathbb{R}^d)\) and it is called weighted Besov-Hölder space with regularity \(s\). Moreover, if \(s \in \mathbb{R}_+ \setminus \mathbb{Z}\), the space \(C^s_{\ell} (\mathbb{R}^d)\) coincides with the Banach space of \(s\)-Hölder-continuous functions.

The relation between weighted Besov spaces is stated in the following result.

Proof. The proof can be found in Theorem 6.7 in [Tri06].\(\Box\)

If \(p = q = 2\), the Besov space \(B_{2, 2, \ell}^s\) coincides with the Sobolev space \(H_{\ell}^s\), i.e. the space of measurable tempered distributions \(f\) with bounded norm

The following theorem allows to extend products between a smooth function and a distribution to elements of Besov spaces.

Proof. See Section 3.3 in [MW17] for Besov spaces with exponential weights. The proof for polynomial weights follows in a similar way.\(\Box\)

In the case of products between a positive measure and an element of a Besov space, the previous result can be improved as follows.

Proof. See, e.g., Lemma 28 in [ADG21].\(\Box\)

The next result is an interpolation estimate for Besov spaces.

Proof. The proof is based on the fact that the complex interpolation of the two spaces \(B^{s_1}_{p_1, q_1, \ell_1} (\mathbb{R}^n)\) and \(B^{s_2}_{p_2, q_2, \ell_2} (\mathbb{R}^n)\) is given by \(B^{s_{\theta}}_{p_{\theta}, q_{\theta}, \ell_{\theta}} (\mathbb{R}^n)\). Such an interpolation is shown in Theorem 6.4.5 in [BL76] for unweighted Besov spaces. The proof for weighted spaces follows from the fact that \(f \in B^s_{p, q, \ell}\) if and only if \(f \cdot \rho_{\ell} \in B^s_{p, q}\) (see Theorem 6.5 in [Tri06]).\(\Box\)

We introduce now the heat kernel and present some of its properties. Let \(P_t = e^{- (- \Delta + m^2) t}\), we consider \(f \in L^r_{\ell_1} (\mathbb{R}, B^s_{p, q, \ell_2} (\mathbb{R}^2))\) and define the heat kernel on \(f\) as

Notice that \(e^{- (- \Delta + m^2)} f (t)\) is a distribution.

More precisely, if \(s > 0\) and if \(f (t, x)\), \((t, x) \in \mathbb{R} \times \mathbb{R}^2\), is a measurable function then

We have the following regularization property for \(e^{- (- \Delta + m^2)}\). Let us remark that all the following results hold also in the case where \(f \in L_{\ell_1}^r ([t_1, t_2], B^s_{p, q, \ell_2} (\mathbb{R}^2))\), where \(- \infty \leqslant t_1 < t_2 \leqslant + \infty \), and the operator \(e^{- (- \Delta + m^2)} f (t)\) is defined as the integral from \(t_1\) to \(t \in [t_1, t_2]\).

Notice that (A.2) states that we are gaining regularity \(\beta_2\) in time and \(2 \beta_1\) in space.

In order to prove Theorem A.6, we need the following result saying that when we apply the heat kernel at time \(t\) we gain \(2 \beta_1\) in space-regularity, but we have to pay with a multiplicative factor of \(t^{- \beta_1}\).

Proof. See Proposition 5 in [MW17].\(\Box\)

We will also need the following lemma saying that giving up some space-regularity we can gain a factor \(t^{\beta}\) on the right-hand side.

Proof. See Proposition 6 in [MW17].\(\Box\)

Proof of Theorem A.6. We give the proof for unweighted Besov spaces, the general case follows the same lines. We use the difference characterization of space-time Besov spaces (see e.g. Theorem 2.36 in [BCD11] or Chapter 2.6.1 in [Tri92]), which yields

First, we prove that the first term on the right-hand side is finite. Write \(\tilde{f} = e^{- (- \Delta + m^2)} f\), we have

Lemma A.7 and Young inequality yield

where the first integral on the last step is finite if and only if \(\beta_1 < 1\).

Consider now the difference term. We have

Now, by Lemma A.7 and Young inequality,

Consider \(\beta_2 < \tilde{\beta} < 1 - \beta_1\), then, by Lemma A.8,

where we used the first part of the proof in the last step and the fact that \(\beta_1 + \tilde{\beta} < 1\). Putting everything together, we get

which gives the result.\(\Box\)

Thanks to the heat kernel, we have another representation for weighted Besov spaces.

Proof. See Section 2.6.4 in [Tri92] for a version of the theorem with a mass-less heat kernel. In particular, such a result differs from the one presented above since the integral with respect to \(t\) in equation (A.3) goes from \(0\) to \(1\) instead of going from zero to \(+ \infty\). This extension is possible thanks to the regularization properties of the heat kernel and the exponential decay thereof, due to the presence of the positive mass \(m > 0\).\(\Box\)

Remark A.10. Under the same hypotheses as in Proposition A.9, if we assume further that \(s > 0\), then the norm \(\| \mathcal{F}^{- 1} (\varphi_0 \mathcal{F} (f)) \|_{L^p_{\ell}}\) appearing in equation (A.3) can be substituted by the weighted \(L^p\)-norm of \(f\), since we have

Namely, for \(s > 0\), equivalence (A.3) becomes

Appendix BStochastic estimates for the Wick exponential

We prove here some stochastic estimates for the term \(: e^{\alpha g_{\varepsilon} \ast X} :\). We need two different kind of estimates. One has to be at the initial time with respect to the Gaussian free field, while the second one needs to estimate the term in some \(L^p\)-space with respect to the variable \(t\). Given \(g_{\varepsilon}\) as in Section 2.2 and \(m > 0\), and \(\mu^{\operatorname{free}}\) being the Gaussian free field with mass \(m\), we define the Wick exponential \(: \exp (\alpha g_{\varepsilon} \ast X) :\) as in equation (2.13).

The previous expression coincides with the standard Wick exponential in the case where we equip the space \(B_X\) with the free field measure \(\mu^{\operatorname{free}}\).

Proof. We report here the proof for the case \(\alpha^2 < 4 \pi\), the general case can be obtained mixing the method presented here and the techniques by [HKK22]. We consider first the case \(r = 2\). We have, for \(K_j =\mathcal{F}^{- 1} (\varphi_j)\),

By translation invariance and orthogonality of Wick polynomials, we can consider

It then suffices to show that, for each \(n \in \mathbb{N}\),

Indeed, for \(| \alpha | < \sqrt{4 \pi}\), we have

We have then to show

for some \(\{ c_n \} \in \ell^1 (\mathbb{N})\).

We want an uniform bound on \(\| \Delta_j : (g_{\varepsilon} \ast X)^n : \|_{L^2}\), which would then give, as \(\varepsilon \rightarrow 0\), the convergence \(\Delta_j : (g_{\varepsilon} \ast X)^n : \rightarrow \Delta_j : X^n :\) almost surely.

Notice that

where

By Lemma 2.2 in [ORW21], we have, for some constant \(C > 0\),

Moreover, if \(x \neq 0\), we have the point-wise limits

Now,

where the multiplicative constant absorbed in the symbol \(\lesssim\) does not depend on \(j\), \(\varepsilon\), and \(n\).

Summing up, we have

Then,

which yields

Therefore, we need

Now, we have

Notice that each term in the sum converges to \(0\) in \(n\) and \(j\). Moreover, we have an uniform bound in \(n\) and \(j\), since

where the term on the right-hand side is summable, and therefore it is in \(\ell^1 (\mathbb{N}^2)\). By Lebesgue dominated convergence theorem, everything converges to zero.

We follow the proof of Theorem 3.8 in [BD21] to address the case \(r > 2\). Let \(r > 2\), take \(\gamma > 1\), and recall hypercontractivity of Gaussian Wick monomials (see e.g. Chapter III in [Üst95]):

We have

Taking \(\kappa\) such that

we conclude the proof in the same way as in the case \(r = 2\).\(\Box\)

We need also a result for the periodic setting. In particular, we prove the convergence of the Wick exponential \(: e^{\alpha Q_{N, M} (g_{\varepsilon} \ast X)} :\) introduced in equation (4.2), as \(N \rightarrow + \infty\)..

Proof. Notice that, if \(Z\) is a Gaussian random variable, then

Therefore, applying the previous reasoning for \(Z = Q_{N, M} (g_{\varepsilon} \ast X)\) and first \(\beta = \alpha\), and then \(\)\(\beta = p \alpha\), then we have

We observe that, if \(X \in C^{- \delta} (\mathbb{T}^2_M)\), then, for every \(x \in \mathbb{T}^2_M\),

Moreover, we have that

and also

These last properties imply that

Taking the norms, we have by translation invariance

and the last term converges to zero by (B.2).\(\Box\)

We now prove convergence in space-time of the Wick exponential.

Proof. The proof follows closely the one of Proposition B.1. See also Theorem 3.2 in [HKK22] and Lemma 2.5 in [HKK21].\(\Box\)

Appendix CEstimates on linearized PDEs

Consider the partial differential equation

where \(B : \mathbb{R}_+ \times \mathbb{R}^2 \times H^1 (\mathbb{R}^2) \rightarrow \mathbb{R}\) is a positive function with compact support with respect to \(z \in \mathbb{R}^2\) independently of \(t \in \mathbb{R}_+\), \(A : B^s_{2, 2, \ell} (\mathbb{R}^2) \rightarrow C^{\infty} (\mathbb{R}^2)\) is a linear and bounded operator which is self-adjoint with respect to the \(L^2 (\mathbb{R}^2)\) Hilbert space structure and which commutes with the Laplacian \(- \Delta\), while \(C : \mathbb{R}_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}\) is a measurable function such that, for any \(t \geqslant 0\), we have

where \(\psi \in L^2 (\mathbb{R}_+, H^1 (\mathbb{R}^2)) \cap L^{\infty} (\mathbb{R}_+, L^2 (\mathbb{R}^2))\).

In this section, we prove some a priori estimates for equation (C.1).

Proof. Multiplying equation (C.1) by \(A \psi (t, z)\) and integrating we have,

Then, exploiting Young's inequality,

Since \(\| A \psi (s, \cdot) \|_{L^2} = \| A^{1 / 2} A^{1 / 2} \psi (s, \cdot) \|_{L^2} \lesssim \| A^{1 / 2} \psi (s, \cdot) \|_{L^2}\), we can reabsorb the last term on the right-hand side and apply Young's convolution inequality to the remaining term to get

This concludes the proof.\(\Box\)

Let us modify the previous result in order to deal with weighted norms. In particular, we consider the case where \(A \psi = g_{\varepsilon} \ast \psi\), where \(g_{\varepsilon}\) is defined as in Section 3, with \(A^{1 / 2} \psi = \tilde{g}_{\varepsilon} \ast \psi\). Let us also recall the definition \(\rho^k_{\ell} (z) = (1 + k | z |^2)^{- \ell / 2}\), for \(z \in \mathbb{R}^2\) and \(k > 0\).

Proof. Instead of multiplying equation (C.1) by \(A \psi (t, z)\), we multiply it by \(A^{1 / 2} (\rho^k_{- \ell} A^{1 / 2} \psi (t, z))\). Proceeding as in the previous proof and noticing that

we have, integrating by parts,

Let us focus on the term

Multiplying and dividing by \(\rho^k_{- \ell} (z)\) inside the integrals gives

By Young's inequality we have

and now we have

This yields

Applying Young's inequality we get

reabsorbing the terms multiplied by \(\sigma_2\) and \(\sigma_3\) respectively, to the left-hand side and noticing that from the proof of Theorem C.1 we also get

we have, renaming the constants and introducing \(\tilde{\sigma}\),

This concludes the proof.\(\Box\)

We now apply a bootstrap argument to the previous result to get the following statement.

Proof. From Theorem C.2, we know that \(A^{1 / 2} \psi\) lives in \(L^2_{\ell'} (\mathbb{R}_+, H_{- \ell}^1 (\mathbb{R}^2)) \cap L^{\infty}_{\ell'} (\mathbb{R}_+, L_{- \ell}^2 (\mathbb{R}^2))\). By interpolation and using the fact that \(H^1_{- \ell} (\mathbb{R}^2) \subset B^0_{p, p, - \ell} (\mathbb{R}^2)\), for every \(p < + \infty\), we have

Applying the heat kernel to the equation (C.1), we have

and therefore, if \(1 / q + 1 / p = 1\) and \(\beta > 0\), we have by Theorem A.6,

On the other hand, for some second degree polynomial \(\mathfrak{P}_2\), we have

This concludes the proof.\(\Box\)

Appendix DProof of Lemma 3.4

We give here the proof of Lemma 3.4. We only show the first part of the result, the second one following in a straightforward way. It is sufficient to prove the following property: for any \(G \in \mathcal{F}\) there exists a sequence of cylinder functions \((G_k)_{k \in \mathbb{N}}\) (here we consider a sequence \((G_{N, M})_{N, M \in \mathbb{N}}\) depending on two parameters) such that \(\mathcal{L}G_k \rightarrow \mathcal{L}G\) point-wise as \(k \rightarrow + \infty\), and we have the uniform bound

for some measurable \(F_G \in L^1 (\mu^{\operatorname{free}})\), e.g. some polynomial of \(X\). And secondly we have to show that \(\mathcal{L}_{\varepsilon} G \rightarrow \mathcal{L}G\) point-wise as \(\varepsilon \rightarrow 0\) with a bound \(| \mathcal{L}_{\varepsilon} G | \leqslant F_G (X)\), for some \(F_G \in L^1 (\mu^{\operatorname{free}})\). Since we take \(\mu \in \mathcal{M}_{B_Y}\), the result follows from Lebesgue's dominated convergence theorem.

Take \(G \in \mathcal{F}\) and \(N, M, \in \mathbb{N}\), and define

where \(f_M = f (\cdot / M)\), \(f : \mathbb{R}^2 \rightarrow [0, 1]\) being a compactly supported smooth function in \(\mathbb{T}^2_1\) which is identically \(1\) in a neighborhood of the origin, and \(Q_{N, M}\) is the operator defined in Section 4. We need to show that

We focus only on the proof of (D.2) and (D.4) since the limit (D.3) follows with similar arguments. We have,

On the other hand, since the integrability parameters in the Besov spaces \(\tilde{B}_X\), \(B_Y\), \(B_{1, 1, - \ell}^{2 - \delta} (\mathbb{R}^2)\), and \(B^{(2 - s) \wedge (\gamma (r - 1)) + \delta}_{l, l, - \ell} (\mathbb{R}^2)\) are finite, the linear operator \(Z \mapsto f_M Q_{N, M} (f_M Z)\), where \(Z \in \mathfrak{B}\), where \(\mathfrak{B}\) is anyone of the Besov spaces listed before, strongly converges to the identity \(\operatorname{id}_{\mathfrak{B}}\) on \(\mathfrak{B}\). Therefore, by continuity of \(\nabla_X G\), we get that the following convergence holds as \(N, M \rightarrow + \infty\):

By strong convergence of the operator \(Z \mapsto f_M Q_{N, M} (f_M Z)\) in \(B_{1, 1, - \ell}^{2 - \delta} (\mathbb{R}^2)\) and the fact that the composition of strongly convergent operators is strongly convergent, we get the limit in (D.2).

Let us now prove (D.4). We have, with the notation \(p_{N, M} = (f_M Q_{N, M} (f_M X), f_M Q_{N, M} (f_M Y))\), and \(p = (X, Y)\),

Let us deal first with the second term of equation (D.5), that is

We have

where we use that \(f_M\), \(Q_{N, M}\), and \(\nabla^2 G (p)\) are self-adjoint operators in \(L^2 (\mathbb{R}^2)\), and \((h^G_n)_{n \in \mathbb{N}}\) is an orthonormal basis of \(L^2 (\mathbb{R}^2)\) of eigenvectors related to the eigenvalues \((\lambda_n)_{n \in \mathbb{N}}\) of the operator \(\nabla^2 G (p)\) (which exists being \(\nabla^2 G (p)\) a compact operator). Since \(\nabla^2 G (p)\) is also a trace-class operator, we have that \((| \lambda_n |)_{n \in \mathbb{N}} \in \ell^1 (\mathbb{N})\). Furthermore, we get

Since we have the strong convergences \(f_M \rightarrow \operatorname{id}_{L^2 (\mathbb{R}^2)}\) as \(M \rightarrow + \infty\), and \(Q_{N, M} \rightarrow \operatorname{id}_{L^2 (\mathbb{T}_M^2)}\) as \(N \rightarrow + \infty\), and so \(\sup_M \| f_M \|_{L (L^2, L^2)}, \sup_{N, M} \| Q_{N, M} \|_{L (L^2, L^2)} < + \infty\), then we have

Thus, we have \((f_M Q_{N, M} f_M)^2 h_n^G \rightarrow h_n^G\) in \(L^2\) as \(N, M \rightarrow + \infty\), and therefore

Then the term (D.6) converges to zero by Lebesgue's dominated convergence theorem.

We now show that the first term in equation (D.5), i.e.

converges to zero. Let \(\mathfrak{M}: H^{\kappa}_{- \ell} (\mathbb{R}^2) \rightarrow L^2 (\mathbb{R}^2)\) be the natural isomorphism between the two spaces, where \(\kappa\) and \(\ell\) are the same parameter as in point iii. of Definition 3.1. The natural identification of \(L^2 (\mathbb{R}^2)\) with its dual, allows us to identify the dual map \(\mathfrak{M}^{\ast}\) of \(\mathfrak{M}\) with the natural isomorphism between \(L^2 (\mathbb{R}^2)\) and \(H^{- \kappa}_{\ell} (\mathbb{R}^2)\). We can then write

and therefore, for two points \(p, p_{N, M} \in \tilde{B}_X \times B_Y\),

which is finite since point iii. in Definition 3.1 holds and the operator \(\iota_{H^{\kappa}_{- \ell} \hookrightarrow L^2} \mathfrak{M}^{- 1} (\mathfrak{M}^{\ast})^{- 1} \iota_{L^2 \hookrightarrow H^{- \kappa}_{\ell}}\) is trace class because \(\kappa > 1\) and \(\ell > 1\) (see Remark 3.2). By continuity of the map \(\nabla^2 G : \tilde{B}_X \times B_Y \rightarrow L (H^{- \kappa}_{\ell} (\mathbb{R}^2), H^{\kappa}_{- \ell} (\mathbb{R}^2))\) and the fact that, by similar arguments as the one exploited to show (D.2), \(p_{N, M} \rightarrow p\) as \(N, M \rightarrow + \infty\), we have that \(\operatorname{tr} (| \nabla^2 G (p) - \nabla^2 G (p_{N, M}) |) \rightarrow 0\) as \(N, M \rightarrow + \infty\).

The argument is then concluded by the following observation: if we take the absolute value of the difference in (D.7), then

which converges to zero since \(\sup_{N, M \in \mathbb{N}} \| f_M Q_{N, M} f_M \|^2\) is finite.

In order to get inequality (D.1), we notice that

where \(F_G\) plays the role of \(f_{\Phi}\) in Definition 3.1, as well as similar inequalities for \(\nabla_Y G\) and \(\operatorname{tr} (\nabla^2 G)\), which is due to the fact that the norm of operator \(f_M Q_{N, M} f_M\) is uniformly bounded in \(N, M \in \mathbb{N}\).

The convergence \(\mathcal{L}_{\varepsilon} G \rightarrow \mathcal{L}G\) as \(\varepsilon \rightarrow 0\) is proved in Section 4.4.3.

Appendix ETechnical results for the resolvent equation

We consider here the system of equations (3.3)–(3.4), and give a proof of Proposition 3.8. For notation simplicity, we will write \(Y\) instead of \(Y^{\varepsilon}\) when no confusion occurs. Moreover, if not explicitly specified, all the appearing parameters are assumed to be taken as in Definition 2.6.

E.1Flow equations

We start with a result about existence and uniqueness of solutions to the system of equations (3.3)–(3.4), that is the first part of the statement in Proposition 3.8.

Proof. The unique solution to equation (3.3) is given by the explicit formula

(see, e.g., Theorem 5.4 in [DZ14]). Showing existence of a solution to equation (3.4) is equivalent to showing existence for

We proceed by a Schaefer's fixed point argument (see Theorem 4 in Section 9.2.2 in [Eva10]) to get the result up to a fixed time \(T > 0\). Consider the map \(\mathcal{J}\) given by, for \(t \in [0, T]\),

We need to show that \(\mathcal{J}_{\cdot} (\cdot, Y_0, X) : A \rightarrow \bar{A}\) is a continuous and bounded map, where \(A\) and \(\bar{A}\) are the convex closed subsets, for \(\kappa > 0\) small enough, \(\delta > 0\), \(\theta \in (\kappa \vee \delta, 1)\), \(\ell > 0\) large enough, consisting of non-negative functions, and such that

Exploiting the compact embedding

given by Besov embedding and Corollary 3 in [Sim87], we can then proceed applying Schaefer's fixed point theorem to get existence for every compact subset of \(\mathbb{R}_+\) of the form \([0, \tau]\), for some \(\tau > 0\). We have

by the regularization property of the convolution with \(g_{\varepsilon}\).

We have to prove continuity. For every \(\tilde{Y}, \tilde{Y}' \in A\), we have

We are left to show uniqueness. Take two solutions \(Y\) and \(Y'\) to equation (3.4). Notice that their difference is given by \(Y - Y' = \tilde{Y} - \tilde{Y}'\) and it satisfies

Introduce a positive function \(h : \mathbb{R} \rightarrow \mathbb{R}\) such that \(h (t) \rightarrow 0\) when \(t \rightarrow - \infty\), and with \(| h' (t) | \leqslant C h (t)\), for some constant \(C > 0\). Multiplying the previous expression by \(h (t) (\tilde{Y}_t - \tilde{Y}'_t)\) and integrating with respect to time and space, we have

For the left-hand side, we get by integration by parts

Notice that the last line is positive. Therefore,

which yields uniqueness.\(\Box\)

Remark E.2. The growth with respect to \(T\) of the norm of the solution \(Y\) to equation (3.4) is polynomial. Indeed, recalling that in the proof of Proposition E.1 we have

we get that the time growth of \(\widetilde{Y_t}\) is determined by the growth with respect to \(s\) of the term

Now, \(\exp (\alpha e^{- (- \Delta + m^2) s} (g_{\varepsilon} \ast Y_0))\) does not increase in time, while \(\exp (\alpha (g_{\varepsilon} \ast \tilde{Y}_s))\) is bounded since \(\tilde{Y}_s \leqslant 0\), and hence the growth with respect to \(s\) is determined only by the remaining term, that is the exponential \(f_{\varepsilon} : \exp (\alpha (g_{\varepsilon} \ast X_s)) :\).

In the next result, we prove continuity of the solutions to equations (3.3)–(3.4) with respect to the initial data.

Proof. As far as \(X\) is concerned, continuity follows from the linearity of its representation (E.1). Let us focus on \(Y\). Consider a sequence \((Y_0^n)_{n \in \mathbb{N}}\) converging to some limit \(Y_0\) in \(B_Y \cup B_{\exp}^{r, \ell}\). Then \((Y_0^n)_{n \in \mathbb{N}}\) is bounded in \(B_Y \cup B_{\exp}^{r, \ell}\), and, by the regularization properties of \(g_{\varepsilon}\), we have that \(\exp (\alpha P_t Y_0^n)\) is uniformly bounded with respect to \(z \in \mathbb{R}^2\), \(t\) and \(n\) on the support of \(g_{\varepsilon}\). From inequality (E.2), it follows that the solution \(\tilde{Y}_t (Y_0^{n_k})\) is bounded in the space \(\bar{A} \subset C^{\theta} ([0, \tau], C^{2 - 2 \theta - \delta}_{- \ell} (\mathbb{R}^2))\) and pre-compact in \(A \subset C^{\theta - \kappa} ([0, \tau], C^{2 - 2 \theta - \delta - \kappa}_{- \ell + \kappa} (\mathbb{R}^2))\). Thus, there exists a subsequence \(n_k\) such that \(\tilde{Y}_t (Y_0^{n_k})\) converges to some \(\tilde{Z}\) in \(A\).

On the other hand, since \(\tilde{Y}_t (Y_0^{n_k})\) solves the fixed-point equation

and since \(\mathcal{J}\) is continuous with respect to all its variables (see the proof of Proposition E.1), we have

By uniqueness of solutions to the fixed point equation, we have that \(\tilde{Y}_t (Y_0) = \tilde{Z}\). Finally, since the solution to equation (3.4) with initial data \(Y_0^{n_k}\) is given by

by the continuity of the heat kernel and of \(\tilde{Y}_t\), we get the thesis.\(\Box\)

E.2Derivatives of the flow

In this section, we denote by \(X\) and \(Y\) the solutions to the system of equations (3.3)–(3.4) and we study their derivatives with respect to the initial data \(X_0\) and \(Y_0\).

E.2.1Existence and equations

We now show point i. in Proposition 3.8.

We have that \(X\) is differentiable and that its derivatives \(\nabla_{X_0} X_t\), \(\nabla^2_{X_0} X_t\), and \(\nabla_{Y_0} X_t\) solve equations (3.5), (3.6), and (3.7), respectively. This is because we have the following explicit representation (cf. equation (E.1)) of the solution

which is linear with respect to the initial data. We immediately get that \(\nabla^2_{X_0} X \equiv 0\) and \(\nabla_{Y_0} X \equiv 0\).

We now focus on the derivatives \(\nabla_{X_0} Y\), \(\nabla_{X_0}^2 Y\), and \(\nabla_{Y_0} Y\) of \(Y\), and show that they exist and satisfy equations (3.8), (3.9), and (3.10), respectively. Furthermore, they are all continuous functions with respect to \(X_0\) and \(Y_0\).

Proof. We only give the proof for \(\nabla_{Y_0} Y\), the other cases follows in a similar way. Consider the approximating equation (3.4), that is

with \(Y_0 \leqslant 0\). We denote the solution to equation (E.3) as \(Y_t (Z)\), in order to stress the initial condition \(Z \in B_Y \cup B_{\exp}^{r, \ell}\). We have the integral representation of the solution \(Y\) to (E.3):

In order to compute the derivative with respect to the initial data, we need to perturb it. Therefore, taking \(\lambda > 0\) and \(h \in B_Y \cup B_{\exp}^{r, \ell}\), and considering the difference, we have

But

since \(\)\(\mathcal{G}_{\varepsilon}\) is differentiable in the direction \(g_{\varepsilon} \ast (Y_s (Y_0 + \lambda h) - Y_s (Y_0))\).

Define

so that we can write

Let us write down an equation for

Notice that

and therefore

Now consider, as in the proof of Proposition E.1,

Then, we have

and such a difference satisfies the following equation

By Theorem C.1, we have then the following bound, for some constant \(K > 0\) and every \(0 < \sigma < 1\):

Consider

Notice that \(\hat{Y}_t^{\lambda} (Y_0)\) is negative and moreover it solves

Recall that \(g_{\varepsilon}\) is such that \(g_{\varepsilon} \ast h \in L^{\infty}_{\operatorname{loc}}\) and \(f_{\varepsilon}\) is compactly supported. Therefore,

We then have the uniform estimate on \(H\) given by, for any \(s > 0\) small enough and \(1 < p < + \infty\),

which is uniform both in \(\lambda\) and \(\xi\).

This gives existence of a limit for \(D_{\lambda, h} Y_t (Y_0)\) as \(\lambda \rightarrow 0\).

We are left to show that the limit satisfies equation (3.8). First, we have to prove that the following limit holds

in some suitable space.

By Lemma E.3, we have that \(Y_t (Y_0 + \lambda h) \rightarrow Y_t (Y_0)\) as \(\lambda \rightarrow 0\) in \(B_Y \cup B_{\exp}^{r, \ell}\), and hence, thanks to the regularization properties of \(g_{\varepsilon}\), \(g_{\varepsilon} \ast Y_t (Y_0 + \lambda h)\) converges to \(g_{\varepsilon} \ast Y_t (Y_0)\) uniformly on the support of \(f_{\varepsilon}\). Thus, we have the weak convergence

and, by the uniform estimates (E.7) and the compact embedding \(B^{- s}_{p, p, - \ell} \hookrightarrow B^{- s'}_{p, p, 0}\), with \(s' > s\) (see Proposition A.2), we get the strong convergence in the space \(B^{- s'}_{p, p, 0}\).

We then have a weak limit for \(D_{\lambda, h} Y_t (Y_0)\) as \(\lambda \rightarrow 0\). Thus, proceeding as in the proof of Lemma E.3, thanks to the a priori estimate (E.6), to the uniform bound (E.7), and to the convergence (E.8), we get that the limit of \(D_{\lambda, h} Y_t (Y_0)\) as \(\lambda \rightarrow 0\) is a solution to equation (3.8).\(\Box\)

E.2.2Properties of the flow derivatives

We prove here some bounds on \(\nabla_{X_0} Y_t (Y_0)\), \(\nabla_{Y_0} Y_t (Y_0)\), and on the trace of \(\nabla^2_{X_0} Y_t (Y_0)\). Let us recall that the sets \(\tilde{B}_Y\) and \(\tilde{B}_X\) are defined as in (3.11).

Proof. We use the same notation as in the proof of Proposition E.4. Recall that

and then, since \(P_t h\) is uniformly bounded in \(\tilde{B}_Y\), in order to prove the result, it suffices to give an estimate on \(D_{\lambda, h} \tilde{Y}_t (Y_0)\) in the space \(C^{\theta}_{\ell'} (\mathbb{R}_+, C_{- \ell}^{2 - 2 \theta - 2 \delta} (\mathbb{R}^2))\).

Let us consider a time-weight \(\rho_{\ell'} (t)\) and a space-weight \(\rho_{- \ell} (x)\) defines as

Then, by Theorem C.2 we have, for some \(0 < \sigma < 1\),

Moreover, applying Corollary C.3 yields

Together with the previous estimate, this yields uniform bounds on the norm \(\| D_{\lambda, h} \tilde{Y}_t (Y_0) \|_{B^{\beta}_{q, q, \ell'} (\mathbb{R}_+, B^{2 - 2 \beta - \delta}_{p, p, - \ell} (\mathbb{R}^2))}\), and choosing \(\beta\) and \(p\) accordingly we then deduce that we have \(D_{\lambda, h} \tilde{Y}_t (Y_0) \in C_{\ell'}^{\delta} (\mathbb{R}_+, C_{- \ell}^{2 - \delta'} (\mathbb{R}^2))\), uniformly in \(\lambda\). Now, letting \(\lambda \rightarrow 0\) yields the result.\(\Box\)

Proof. The proof is similar to the one of Proposition E.5, the only difference is that here the initial data of \(\nabla_{X_0} Y_t\) is zero and therefore we do not need to subtract it before doing the estimates in Theorem C.2 and in Corollary C.3.\(\Box\)

We now deal with the trace term appearing in the definition of the operator \(\mathcal{L}\).

Proof. We suppose that all the computations involving equations (3.9) and (3.10) make sense since a rigorous proof of this fact can be given in a similar way as in the proof of Proposition E.5.

Let \(\delta_{\cdot}\) be the Dirac delta distribution, and consider the map, for \(X_0 \in B_X\) and \(t \in \mathbb{R}_+\),

Hereafter, we drop the dependence on \(Y_0\) in the derivatives of \(Y\). Let us then evaluate equations (3.9) and (3.10) at \(\delta_z (\cdot)\), and \((h, \delta_z (\cdot))\), respectively, to get

and

Using similar methods as in the proofs of Proposition E.4 and Proposition E.5, it is possible to prove that \(\nabla_{X_0} Y_t (\delta_z)\) and \(\nabla^2_{X_0} Y_t (h, \delta_z)\) are differentiable infinitely many times with respect to \(z\) and their derivatives with respect to \(z\) for any multi-index \(\beta\) satisfy

Furthermore, by equations (E.9) and (E.10), we get

and

Exploiting Theorem C.2 applied to equation (E.11), we get

By a bootstrap argument, i.e. applying Corollary C.3, we can conclude that

where \(q_s\) is an \(s\)-dependent polynomial, because of the relation of the heat kernel with the support of \(f_{\varepsilon}\).

By a similar method as above, we get the following estimate concerning the second derivative

where \(\tilde{q}_s\) is another \(s\)-dependent polynomial. Taking \(s\) large enough, we have

This proves that the map is linear with respect to \(h\) as a map from \(L^2\), moreover, if \(s \in \mathbb{N}\) and fixing \(z'\) and \(h\), the norm with respect to \(z\) is given by

We are left to show that the last integral is finite. Recall

Since we have to show that the integral is finite for some polynomial, we multiply (and divide) the heat kernel by some weight \(\rho_{- 2 \ell}\), to get

By inequality (6.2) in Section 6 of [Tri06], we have, for some \(\kappa > 0\),

Therefore,

By the compact support of \(g_{\varepsilon}\), we have

where \(W\) is some two-dimensional Brownian motion. Thus, if we take \(\kappa\) large enough, we get

Then,

which is finite, since \(f_{\varepsilon}\) is compactly supported.\(\Box\)

References