The aim of this paper is to develop further the variational approach to Euclidean quantum fields initiated in \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\) by studying the infinite volume limit of the variational description and some of its consequences. For simplicity we will focus on the two dimensional setting. We will highlight some unsolved issues which would have to be worked out in this simpler setting before moving to more challenging models. The infinite volume limit in the \(\varphi^4_3\) theory considered in \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\) can be attacked along the lines suggested here.

The key tool is a novel kind of stochastic equations for Euclidean fields which involve an additional one dimensional parameter. This parameter already appeared in \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\) linked to a continuous scale decomposition of the relevant random fields. Here the additional parameter \(t\) can be chosen to take values in \([0, 1]\) with no particular association to the scale of fluctuations thanks to the milder irregularity of the two dimensional Gaussian free field and to the simpler renormalization (i.e. Wick ordering) needed. In a general perspective one can interpret the variational method and the associated stochastic equations as a way to stochastically quantize a given field theory, alternative to the classical approach of Parisi–Wu via a Langevin equation [32] and to the elliptic stochastic quantization recently introduced in [2, 1] via the supersymmetric dimensional reduction of Parisi–Sourlas [31].

The variational method we introduced in \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\) has been recently used in various contexts to provide novel results: Bringmann used it to construct singular Gibbs measures [11] and study their invariance under dispersive dynamics [12], Chandra, Gunaratnam and Weber used it to show the existence of phase transition in the \(\varphi^4_3\) model [13], Oh, Okamoto and Tolomeo [30] to prove existence and non-existence of certain Gibbs measures. In [5] we provide another approach to describe the most singular part of the \(\varphi^4_3\) measure by approximating the optimal drift of the variational description. The PhD thesis of the first author [4] describes also how to study the infinite volume limit of the Sine–Gordon model via the variational method including a full verification of the Osterwalder–Schrader axioms.

Similar ideas relying on a coupling between the Gaussian free field and an interacting field have been used by Bauerschmidt and Hofstetter to control the maximum of the Sine–Gordon model [6]. More generally, mapping a Brownian motion into a measure of interest has revealed itself an useful method also in the context of functional inequalities, see e.g. the work of Borell [8], Lehec [28] and the recent paper of Mikulincer and Shenfeld [29] on the “Brownian transport map”.

Let us now give an informal and heuristic picture of the results of this paper. The basic object of interest is a two dimensional Euclidean quantum field [34, 17], namely a probability measure \(\theta\) on the Schwarz space \(\mathcal{S}' (\mathbb{R}^2)\) of tempered distributions given by the expression

where \(\theta^0\) is the Gaussian free field (GFF) on \(\mathbb{R}^2\) with covariance \((m^2 - \Delta)^{- 1}\) and \(V\) is a local interaction term given by

with \(\xi : \mathbb{R}^2 \rightarrow \mathbb{R}\) a compactly supported function to make the spatial integral finite and \(p\) is either a polynomial of even degree or an exponential function.

We interpret the GFF \(\varphi\) as the terminal value of a Brownian martingale \((W_t)_{t \in [0, 1]}\) with values in \(\mathcal{S}' (\mathbb{R}^2)\) and covariance given by

If we denote by \(\mathbb{P}\) the law on the path space \(C ([0, 1], \mathcal{S}' (\mathbb{R}^2))\) of this Brownian motion. The measure \(\theta\) can be then interpreted as the law of \(W_1\) under the Gibbs measure

It is well known that the measure \(\mathbb{Q}^{\xi}\) is the minimizer of a Gibbs variational problem and that

where \(\mathbb{E}_{\mathbb{R}}\) denotes the expectation with respect to a probability measure \(\mathbb{R}\) on the path space \(C ([0, 1], \mathcal{S}' (\mathbb{R}^2))\) and \(H (\mathbb{R}|\mathbb{P})\) the relative entropy of \(\mathbb{R}\) with respect to \(\mathbb{P}\). A formula of Boué–Dupuis [9, 35] allows to rewrite the variational problem as an optimal control problem, namely

where the infimum is now taken over all processes \((Z_t)_{t \in [0, 1]}\) with values in \(H^1 (\mathbb{R}^2)\) and adapted to the filtration of \(W\) and

is a bilinear form giving the “energy” of the drift \(Z\). Here \(\dot{Z}_s\) denotes the derivative of \(Z_s\) with respect to the parameter \(s \in [0, 1]\). As we show in this paper, the meaning of an optimal drift \(Z\) is that of providing the law of the “interacting field” via the equality

valid for any minimizer \(Z\) of the problem (1) and in particular

The existence of such minimizers will be guaranteed, in the rigorous discussion below, by suitably relaxing the variational problem into a problem on the space of probability measures. For pedagogical reasons let us for the moment continue our heuristic discussion ignoring these fine points.

The variational description (1) looses meaning in infinite volume, i.e. when \(\xi \equiv 1\), since the cost functional is always infinite. However the main idea in this paper is to note that any minimizer \(Z\) of (1) satisfy the Euler–Lagrange equation (first variation)

for all “test controls” \(K\), where \(\nabla V (W_1 + Z_1) \cdot K_1\) denotes the derivative of \(\varphi \mapsto V (\varphi)\) at \(W_1 + Z_1\) and in the direction of \(K_1\). In the following we will refer to the equation (3) as a weak forward–backwards stochastic (partial) differential equation (wFBSDE). The reason for this name derives from observing that (3) can be formally rewritten as

and since this holds for any nice adapted control \(K\) it implies that

which is a kind of elliptic equation for the optimal control \((Z_s)_{s \in [0, 1]}\). It can be cast into a system of a forward SDE for \(Z\) and a backward SDE which computes the conditional expectation. In this paper we will not exploit further this specific interpretation of (3) as a FBSDE so it is not necessary to make it more precise. We would like to stress however that our wFBSDE is a new kind of stochastic quantisation for Euclidean QFTs. Indeed, equation (3) makes sense also in infinite volume and can be used, very much like the weak formulation of an elliptic problem, to get estimates on the minimizer \(Z\), e.g. by testing with \(K = \rho Z\) where \(\rho : \mathbb{R}^2 \rightarrow \mathbb{R}_+\) is some weight function decaying at spatial infinity. In particular we will obtain estimates of the form

uniformly in the cutoff \(\xi\). This estimate together with the identification (2) is at the core of the proof of tightness of the family of measures \((\mathbb{Q}^{\xi})_{\xi}\) with \(\xi \rightarrow 1\). Any accumulation point \(\mathbb{Q}\) represents an infinite volume Euclidean field, moreover the corresponding drift \(Z\) gives, via the formula (2) an explicit coupling of \(\mathbb{Q}\) and the GFF given by \(\mathbb{P}\). The point of view which we advocate in this paper is that the wFBSDE equation (3) should provide a complete control over the drift \(Z\) describing the infinite volume Euclidean field, for which a naive variational description is meaningless. Moreover intuitively we should be able to introduce a variational description of the measure \(\mathbb{Q}\) with drift \(Z\) via a renormalized BD formula of the form

with \(X_s = W_s + Z_s\) being the “infinite volume interacting field” with potential \(V\) and \(f (\varphi)\) a test function which depends only locally on the field \(\varphi\) and where the infimum is taken among all drifts \(K\) which satisfy suitable decay at infinity. Given this decay one expects that the quantities \(V (X_1 + K_1) - V (X_1)\) and \(\mathcal{E} (Z + K) -\mathcal{E} (Z)\) are finite even if each summand is infinite.

Leaving this heuristic account of the ideas investigated in this paper, we describe now the plan for the rigorous proofs. We will start in Section 1 by introducing the general setting for the variational description of two dimensional Euclidean quantum fields as outlined above. We prove certain useful bounds and also that the minimizers of this variational problem describe in a precise sense the Euclidean field and moreover that they satisfy wFBSDEs.

As a first example, we address in Section 2 the problem of the infinite volume limit of the \(\varphi^4_2\) measure \(\theta_{\Lambda}\) on a bounded domain \(\Lambda \subseteq \mathbb{R}^2\). From the wFBSDE we will derive a-priori bounds for minimizers strong enough to give tightness of the family \((\theta_{\Lambda})_{\Lambda}\) as \(\Lambda \nearrow \mathbb{R}^2\). Unfortunately, establishing rigorously the form (5) of the limiting variational problem requires certain locality properties of the solutions to the wFBSDE (3) in infinite volume. This is usually possible when \(V\) is convex. Ultraviolet renormalization in \(\varphi^4_2\) however spoils any “easy” convexity of the potential and it is not clear how to gain sufficient control of (3) in this case. Similar issues appear in other methods of stochastic quantization as soon as the infinite volume limit is considered for \(P (\varphi)_2\) Euclidean quantum fields [18, 19, 1]. Despite these difficulties, in Section 3 we show that the wFBSDE is still a useful tool which can be used effectively to prove the large deviations of the \(\varphi^4_2\) measure in the semiclassical limits. Let us remark that Lacoin Rhodes and Vargas obtained large deviations results in the case of the Liouville model in [26, 27].

Finally, in Section 4, we discuss the case of the exponential interaction, that is the \(\exp (\beta \varphi)_2\) model (or Høegh–Krohn model) introduced in [21]. See e.g. [1] for a discussion of its stochastic quantization in the elliptic setting. For this model we repeat the construction of the variational description of the infinite volume measure. At variance with what happens in \(P (\varphi)_2\) theories, renormalization does not spoil the convexity properties of the potential and therefore the study of the related wFBSDE can go further and in particular we will be able to prove the necessary locality of the perturbed equation. This in turn will allow us to prove the \(\Gamma\)-convergence of the variational problem along with the infinite volume limit and establish eq. (5) in its rigorous form. As a result of this analysis, we are able to obtain a concise and well-behaved description of an infinite volume Gibbs measure related to an Euclidean quantum field.

Acknowledgments.

Notation.

where \(\mathcal{F}\) is the Fourier transform on \(\mathbb{R}^2\).

All along the paper we need to consider Banach subspaces of \(\mathcal{S}' (\mathbb{R}^2)\) associated with a positive spatial weight \(\rho\). Let \(\rho : \mathbb{R}^2 \rightarrow \mathbb{R}_{> 0}\) a smooth positive function such that for some sufficiently small \(\varepsilon > 0\) we have

The number \(\varepsilon\) can depend on various parameters but not on the volume cutoffs we introduce to regularize the Euclidean QFTs. For some \(\alpha > 0\), \(s \in \mathbb{R}\), \(p, q \in [1, \infty]\), we introduce the usual weighted Sobolev spaces \(W^{s, p} (\rho^{\alpha})\) and weighted Besov spaces \(B_{p, q}^s (\rho^{\alpha})\) with norms

and let \(H^s (\rho^{\alpha}) = W^{1, 2} (\rho^{\alpha})\) and \(C^s (\rho^{\alpha}) = B_{\infty, \infty}^s (\rho^{\alpha})\). Here \(B_{p, q}^s\) is the unweighted Besov norm given by

where the Littlewood–Paley operators \((\Delta_i)_{i \geqslant - 1}\) are defined by \(\Delta_i = \psi (\mathrm{D} / 2^i)\) for \(i \geqslant 0\) and some function \(\psi : \mathbb{R}^2 \rightarrow \mathbb{R}_{\geqslant 0}\) supported in an annulus centered at the origin and \(\Delta_{- 1} = \tilde{\psi} (\mathrm{D})\) for \(\tilde{\psi} : \mathbb{R}^2 \rightarrow \mathbb{R}_{\geqslant 0}\) supported on a ball centered at the origin and such that \(\sum_{i \geqslant - 1} \Delta_i = 1\).

In general we avoid to indicate the weight when dealing with the unweighted (\(\alpha = 0\)) version of the above spaces.

We denote by \(\mathcal{P} (X)\) the set of Borel measures on the topological space \(X\), for \(P \in \mathcal{P} (X)\) we denote \(\mathbb{E}_P\) the corresponding expectation and for any random variable \(H : X \rightarrow Y\) on the topological space \(Y\) we let \(\operatorname{Law}_P (H) \in \mathcal{P} (Y)\) the corresponding law.

1A variational setting for 2d Euclidean fields

Consider the Gibbs measure

where \(\theta^0\) is the law of the Gaussian field \(\phi\) (GFF) on \(\mathbb{R}^2\) with covariance \((m^2 - \Delta)^{- 1}\) for some mass \(m > 0\) (fixed throughout the paper) and where \(\Delta\) denotes the Laplacian on \(\mathbb{R}^2\). The function \(V : \mathcal{S}' (\mathbb{R}^2) \rightarrow \mathbb{R}\) represent an interaction potential and for now we will only assume that it is measurable and tame with respect to \(\theta^0\) according to the following definition.

Tameness has been introduced by Üstunel [35] as a convenient condition for the validity of the Boué–Dupuis variational formula for Brownian functionals. It has been recently improved by Hariya and Watanabe [20] to allow for \(p = q = 1\). To introduce this formula we need to enlarge our probability space and realize the GFF as a marginal of a Brownian motion, following the approach initiated in \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\). Here we are able to use a simpler embedding thanks to the milder nature of the Gaussian free field in two dimensions compared to its three dimensional counterpart.

Let \(\mathbb{P}\) be the law of a cylindrical Brownian motion \((X_t)_{t \in [0, 1]}\) on \(L^2 (\mathbb{R}^2)\). It can be realized on \(C ([0, 1], C^{- 1 - \varepsilon} (\rho^{\varepsilon}))\) where \(\varepsilon > 0\) can be taken arbitrarily small. Let

and note that indeed \(\theta^0 =\operatorname{Law}_{\mathbb{P}} (W_1)\). We will consider \(W\) as a random variable in \(\mathfrak{S}_0 := C ([0, 1], C^{- \varepsilon} (\rho^{\varepsilon}))\).

Let \(\mathfrak{H} (\rho^{\alpha}) = H_t^1 H_x^1 (\rho^{\alpha})\), \(\mathfrak{H} = \mathfrak{H} (\rho^0)\). Denote with \(\mathfrak{H}^a (\rho^{\alpha})\) the space of all processes \(u : \mathfrak{S}_0 \times [0, 1] \rightarrow \mathbb{R}\) progressively measurable with respect to the filtration generated by \((W_t)_{t \in [0, 1]}\), starting from \(0\) and equipped with the norm of \(L^2 (\mathbb{P}, \mathfrak{H} (\rho^{\alpha}))\). Moreover define \(\mathcal{E}: \mathfrak{H}^a \rightarrow L^0 (\mathbb{P})\) as

where \(\dot{Z}\) denotes the time-derivative of \(Z\). With these definitions, the Boué–Dupuis formula reads as follows.

Proof. Theorem 7 from Üstunel [35] specialized in our context (see also \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\)) gives

where \(\mathbb{H}^a\) is the space of all progressively measurable processes starting from \(0\) equipped with the norm of \(L^2 (\mathbb{P}, L_t^2 L^2_x)\) and

Observing that \(u_t = (m^2 - \Delta)^{1 / 2} \dot{Z}_t\) we obtain the claim.\(\Box\)

Note that the functional \(G (W_{\cdot} + Z_{\cdot})\) makes sense for any \(Z \in \mathfrak{H}^a\) since by Theorem 2 from Üstunel [35] we have that the push-forward measure \(\mathbb{Q}= (W_{\cdot} + Z_{\cdot})_{\ast} \mathbb{P}\) has finite relative entropy with respect to \(\mathbb{P}\) and therefore is absolutely continuous, i.e. \(\mathbb{Q} \ll \mathbb{P}\).

We call a random variable with values in \(\mathfrak{H}^a\) a control (or drift) and we will understand that \(W = W_1\) and \(Z = Z_1\) whenever this does not lead to confusion.

To simplify certain technical considerations later is convenient to augment the process \(W\) with some additional informations. Consider a Polish space \(\mathfrak{S}\) endowed with a continuous projection \(\pi_W : \mathfrak{S} \rightarrow \mathfrak{S}_0\) and let \(\mathbb{W}: \mathfrak{S}_0 \rightarrow \mathfrak{S}\) a measurable map such that \(\pi_W \mathbb{W}= W\), i.e. \(\mathfrak{S} \xrightarrow{W} \mathfrak{S}_0\) is a fibered topological space and \(\mathbb{W}\) a measurable section. We will understand \(\mathbb{W}\) as an enhanced GFF. The kind of enhancements needed will be different in different models. We will always consider the space \(\mathfrak{S}\) endowed with the measure given by the law of \(\mathbb{W}\) under \(\mathbb{P}\). A functional on \(\mathfrak{S}\) is tame if it is tame for this measure.

For any measurable \(f : \mathcal{S}' (\mathbb{R}^2) \rightarrow \mathbb{R}\) bounded below and in \(L^2 (\theta^0)\) (not necessarily linear) define \(\mathcal{W}^V (f)\) as

As a direct consequence of Lemma 2, the functional \(\mathcal{W}^V\) has the variational representation

for any \(f\) such that \(f + V\) is tame and where for a tame functional \(G\) on \(\mathfrak{S}_0\) we let

To proceed we want to find a minimizer of \(F^{f + V}\). It is however not clear if we can find one in \(\mathfrak{H}^a\). Following the approach we used in \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\), we relax our problem and instead minimize over the law of the pair \((W, Z)\) with \(W\) fixed and \(Z\) varying over \(\mathfrak{H}^a\). In order to ensure compactness we need to complete the space of trial laws for a suitable topology for which our target functional is coercive.

Let \(\mathfrak{H}_w (\rho^{\alpha})\) to be \(\mathfrak{H} (\rho^{\alpha})\) equipped with the weak topology and let \(\widehat{\mathfrak{H}} := \mathfrak{H} (\rho^{1 / 2})\), \(\widehat{\mathfrak{H}}_w := \mathfrak{H}_w (\rho^{1 / 2})\) and \(\widehat{\mathfrak{H}}^a := \mathfrak{H}^a (\rho^{1 / 2})\). With a slight abuse of notation we denote \((\mathbb{W}, Z)\) the canonical coordinates on \(\mathfrak{S} \times \widehat{\mathfrak{H}}_w\).

We will need to realize couplings of two elements of \(\overline{\mathcal{X}}\). These are constructed by the next lemma which needs the following preliminary definition of the space hosting the coupling. Denote the canonical coordinates on \(\mathfrak{S} \times \widehat{\mathfrak{H}}_w \times \widehat{\mathfrak{H}}_w\) by \((\mathbb{W}, Z, K)\).

Proof. Since \(\mu, \nu\) are in \(\overline{\mathcal{X}}\) we can find sequences \(Z^n, \bar{Z}^n \in \widehat{\mathfrak{H}}^a\) such that \(\mu_n : =\operatorname{Law}_{\mathbb{P}} (\mathbb{W}, Z^n) \rightarrow \mu\) and \(\nu_n := \operatorname{Law}_{\mathbb{P}} (\mathbb{W}, \bar{Z}^n) \rightarrow \nu\) weakly in \(\widehat{\mathfrak{H}}_w\) and furthermore \(\sup_n \mathbb{E} [\| Z^n \|_{\widehat{\mathfrak{H}}}^2 + \| \bar{Z}^n \|_{\widehat{\mathfrak{H}}}^2] < \infty\). Now define

and set

We have that \(\sup_n \mathbb{E} [\| K^n \|^2_{\mathfrak{\hat{H}}}] \leqslant \sup_n \mathbb{E} [\| Z^n \|_{\mathfrak{\hat{H}}}^2 + \| \bar{Z}^n \|_{\mathfrak{\hat{H}}}^2] < \infty\) so \(\upsilon^n\) is tight on \(\)\(\mathfrak{S} \times \mathfrak{\hat{H}}_w \times \mathfrak{\hat{H}}_w\). Taking a convergent subsequence (not relabeled) we can assume that \(\upsilon^n \rightarrow \upsilon\). We then have

which proves the claim. \(\Box\)

On functionals on \(\bar{\mathcal{X}}\) we can define first and second order derivatives as follows. Let \(\mu \in \bar{\mathcal{X}}\) and let \(K : \mathfrak{S} \times \mathfrak{\hat{H}}_w \rightarrow \mathfrak{\hat{H}}_w\) a measurable process adapted to the filtration generated by \((\mathbb{W}_t, Z_t)_{t \in [0, 1]}\), continuous in the second component and such that \(K \in L^2 (\mu, \mathfrak{\hat{H}})\). We call such \(K\) an admissible control for \(\mu\). For all \(\sigma \in \mathbb{R}\) let \(\mu^{\sigma K}\) the law of \((\mathbb{W}, Z + \sigma K)\) under \(\mu\). We have \(\mu^{\sigma K} \in \bar{\mathcal{X}}\). A functional \(\check{F} : \bar{\mathcal{X}} \rightarrow \mathbb{R}\) is differentiable in \(\bar{\mathcal{X}}\) in the direction of the admissible control \(K\) if the map \(\sigma \mapsto \check{F} (\mu^{\sigma K})\) is differentiable in \(\sigma = 0\) and in this case we define

Second order derivatives can be similarly defined by taking two admissible controls \(K, K'\) and letting

whenever \((\sigma, \sigma') \mapsto \check{F} (\mu^{\sigma K + \sigma' K})\) is \(C^2\) in \((\sigma, \sigma') = 0\).

We are ready to relax the variational problem and prove an alternative representation of \(\mathcal{W}_{\Lambda} (f)\). For a tame \(G : \mathfrak{S} \rightarrow \mathbb{R}\) define the functional \(\check{F}^G : \bar{\mathcal{X}} \rightarrow \mathbb{R}\) by

The following lemma is an obvious consequence of this relaxed variational setting.

Proof. The equality \(\inf_{Z \in \mathfrak{H}^a} F^G (Z) = \inf_{Z \in \widehat{\mathfrak{H}}^a} F^G (Z)\) follows from the property (8) of the admissible functional \(\check{F}^G\) since \(\mathfrak{H}^a\) can be viewed as a subspace of \(\overline{\mathcal{X}}\) via the map \(u \mapsto \operatorname{Law} (\mathbb{W}, u)\). The equality \(\inf_{Z \in \widehat{\mathfrak{H}}^a} F^G (Z) = \inf_{\mu \in \mathcal{X}} \check{F}^G (\mu)\) follows directly from the definition of \(\mathcal{X}\). We have that \(\inf_{\mu \in \overline{\mathcal{X}}} \check{F}^G (\mu) = \min_{\mu \in \overline{\mathcal{X}}} \check{F}^G (\mu)\) again from the admissibility of \(G\) which ensures that sequences \((\mu_n)_n \subseteq \mathcal{X}\) of approximate minimizers of \(\check{F}^G\) converge in \(\overline{\mathcal{X}}\) and that any limit \(\mu^G\) is a minimizer. The first and second order conditions are then consequences of the differentiability of \(\check{F}^G\). It remains to show

We fist show this statement for bounded \(G\). We want to show that for any \(\mu\) in X^¯ any \(\delta > 0\) there exists \(\mu_n \in \mathcal{X}\)

So given \(\mu \in \overline{\mathcal{X}}\) we take a sequence \(\mu_n \in \mathcal{X}\) such that \(\mu_n \rightarrow \mu\). We define \(\mu^N_n\) to be

Then \(\mu_n^N = \mu_n\) on the event that \(\mathcal{E} (Z) \leqslant N\). Select a subsequence such that \(\mu_n^N \rightarrow \upsilon^N\) in \(\mathcal{X}\). Define \(\theta^{\delta}_{\ast} \nu =\operatorname{Law}_{\nu} (W, \theta^{\delta} \ast Z)\) where \(\theta^{\delta}\) is a sequence of standard mollifiers. Now by weak convergence

since the functional inside the expectation is continuous and bounded on the joint support of the measures \(\theta^{\delta}_{\ast} \mu^N_n\). We have to show that

First we establish that

Since \(\mathcal{E} (\theta^{\delta} \ast Z) \leqslant \mathcal{E} (Z)\) and \(G\) is continuous on \(\mathfrak{S} \times \widehat{\mathfrak{H}}_w\) we have

Now we have to show that

Since \(\upsilon^N\) coincides with \(\mu\) on any open set contained in \(\mathcal{E} (Z) \leqslant N\) we have

For the energy

Now

We now remove the restriction that \(G\) is bounded. To this end let \(G^N = (G \wedge N) \vee (- N)\). Then we have established that already that

Now let \(\nu\) be a minimizer of \(\check{F}^G\) in \(\mathcal{X}\)^¯ (existence of this was established above). Then

Where we have used dominated convergence.\(\Box\)

Remark 9. Note that

If \(G\) is differentiable in the direction of \(H^1\) we have

If \(G\) is twice differentiable we have also

The consequences of the existence of this minimizer are clarified by the following lemmas.

Proof. If \(f + V\) is admissible we have proven that

Eq. (11) implies in turn

and since

for any \(g\) (in particular admissible \(g\)), we obtain eq. (10).\(\Box\)

Proof. The admissibility of \(f + V\) is easy to obtain given that \(f\) is bounded and continuous and \(V\) is admissible. It is clear that \(\alpha \mapsto \mathcal{W}^V (\alpha f)\) is differentiable for bounded \(f\), we need only to establish the representation of this derivative via the variational minimizer \(\mu^{\alpha f}\). For any \(\gamma \geqslant 0\) we have, by direct computation

and

By differentiability of \(\mathcal{W}^V (\alpha f)\) we have

and

from which we can conclude.\(\Box\)

There are several interesting consequences of this formula. First of all, integrating (12) in \(\alpha\), we have

which is a representation of \(\mathcal{W}^V (f)\) alternative to the variational formula. On the other hand, taking \(\alpha = 0\) in (12) we obtain

This last formula naturally calls for a variational description of the average \(\mathbb{E}_{\mu^V} [e^{- f (W_1 + Z_1)}]\) with respect to the base measure \(\mu^V\). This extension of the BD formula to the interacting field \(W + Z\) is the subject of the next lemma.

Let \(\mathcal{E} (Z, K)\) the symmetric bilinear form associated to \(\mathcal{E} (Z)\), in such a way that

Proof. Recall that

where in the last line we used Lemma 5 to justify the change of variables in the minimization problem. Note that for any \(\upsilon \in \overline{\mathcal{Y}} (\mu^V)\) we have that

so

The fact that \(\check{H}^V (\upsilon) \geqslant 0\) iff \(\operatorname{Law}_{\upsilon} (W, Z) = \nu\) is a minimizer of \(\check{F}^V\) follows from its definition, indeed note that

If \(V\) is differentiable in the direction of \(H^1\) we have that for any minimizer \(\nu\) of \(\check{F}^V\) and any \(\upsilon \in \bar{\mathcal{Y}} (\nu)\):

which allow also to simplify the expression of \(\check{H}^V\).\(\Box\)

Remark 14. The interest of this formulation is two fold:

2The quartic interaction

In this section we will consider the above general setting in the case where the interaction potential \(V\) is given by

where \(\Lambda \subseteq \mathbb{R}^2\) is a measurable bounded set (of sufficient regularity, see below) and \(\mathopen{\unicode{x27E6}} \phi^n \mathclose{\unicode{x27E7}}\) denotes the \(n\)-th Wick power of the Gaussian free field \(\phi\) [17] and \(\lambda > 0\) an arbitrary fixed constant. Let \(\mathcal{W}_{\Lambda} := \mathcal{W}^{V_{\Lambda}}\) and \(\theta_{\Lambda} := \theta^{V_{\Lambda}}\). The measure \(\theta_{\Lambda}\) on \(\mathcal{S}' (\mathbb{R}^2)\) is the \(\varphi^4_2\) Euclidean quantum field with a finite volume cutoff.

We will prove that for a sufficiently large class of functions \(f\) the family \((\mathcal{W}_{\Lambda} (f))_{\Lambda}\) is uniformly bounded as \(\Lambda \nearrow \mathbb{R}^2\). This implies that the family of measures \((\theta_{\Lambda})_{\Lambda}\) is tight on \(\mathcal{S}' (\mathbb{R}^2)\). We will describe any accumulation point via a formula for its Laplace transform involving the wFBSDE obtained in the previous section.

Remark 15. Let us note that all the discussion in this section can be extended to general \(P (\varphi)_2\) models with a positive coefficient in the highest order polynomial. Moreover also \(O (N)\) invariant vector models could be handled in similar ways. Convexity of the classical potential (i.e. before Wick ordering) is only needed in Section 3 below to prove large deviations.

We will need the following enhancement \(\mathbb{W}\) of the GFF \(W\). Let

where here \(\mathopen{\unicode{x27E6}} \cdot \mathclose{\unicode{x27E7}}\) denotes the Wick powers with respect to the Gaussian field \(W\). Then

for all \(p \geqslant 1\), where \(\mathfrak{S} := C ([0, 1], (C^{- \delta} (\rho^{- 1 / 4}))^3)\) for a fixed \(\delta > 0\) which can be chosen as small as we like. By abuse of notation we will denote also \(\mathbb{W}= (W, \mathbb{W}^2, \mathbb{W}^3)\) the canonical process on \(\mathfrak{S}\).

In the following we will always consider a family of \((\Lambda)_{| \Lambda | \in \mathbb{R}}\) given by \(\Lambda = | \Lambda |^{1 / 2} S\) for some fixed set \(S\) of volume \(1\) with \(C^1\)-boundary. This allow to prove that \(\mathbb{1}_{\Lambda}\) is sufficiently regular to make sense of quantities like \(\int_{\Lambda} \mathopen{\unicode{x27E6}} \phi^4 \mathclose{\unicode{x27E7}}\), \(\int_{\Lambda} \mathopen{\unicode{x27E6}} W^4_1 \mathclose{\unicode{x27E7}}\) and in general of products with Wick monomials of the GFF.

Proof. We have that \(\nabla \mathbb{1}_{\Lambda} = \nu \mathrm{d} \sigma\) where \(\mathrm{d} \sigma\) is the surface measure on \(\Lambda\) and \(\nu\) is the outer unit normal. It is known that \(\| f \|_{B_{p, p}^s (\rho)} \lesssim \| f \|_{B_{p, p}^s (\rho)} + \| \nabla f \|_{B_{p, p}^{s - 1} (\rho)}\) (see e.g [19], Lemma A.5). This implies that for any \(s < 1\), \(\sup_{| \Lambda |} \| \mathbb{1}_{\Lambda} \|_{B_{1, 1}^s (\rho^{\iota})} < \infty\). Clearly \(\sup_{| \Lambda |} \| \mathbb{1}_{\Lambda} \|_{L^{\infty} (\rho^{\iota})} < \infty\) and by Besov space interpolation (see e.g. [19] Lemma A.3)

\(\| \mathbb{1}_{\Lambda} \|_{L^{\infty} (\rho^{\iota})}\)where

Proof. It is known that

for any \(q > 0\) (see e.g [17] or \([ \hyperref[bib-barashkov_gubinelli_variational]{3}]\)) and

for some \(\delta > 0\), which shows that \(F\) is a tame functional. Now it remains to prove the equality in (15). For this it is enough to show that for \(Z\) such that \(\| Z \|_{H^1 (\rho^{1 / 2})} < \infty\) \(\mu\)-almost surely, we have

To see this, recall that if \(C^{\infty}_c \ni \psi_{\varepsilon} \rightarrow \delta_0\), is a standard mollifier, \(W^{\varepsilon} = \psi_{\varepsilon} \ast W\) and \(a_{\varepsilon} = 6\mathbb{E} [(W^{\varepsilon}_1 (0))^2]\), \(b_{\varepsilon} = 3\mathbb{E} [(W^{\varepsilon}_1 (0))^2]^2\)

where we have denoted \(Z^{\varepsilon}_1 = \psi_{\varepsilon} \ast Z_1\). Now it is well known that almost surely in \(C^{- \delta} (\rho^{\delta})\) we have \(W^{\varepsilon}_1 \rightarrow W_1\) and

Furthermore \(Z^{\varepsilon} \rightarrow Z\) in \(H^1\) which implies \((Z^{\varepsilon}_1)^2 \rightarrow Z^2_1, (Z^{\varepsilon}_1)^3 \rightarrow (Z^{\varepsilon}_1)^3\) in \(W^{2 \delta, 1}\) for some \(\delta > 0\), which gives the statement. From this calculation it also follows that \((W, Z) \rightarrow G (W + Z)\) is continuous on \(\mathfrak{S} \times \mathfrak{H}_w\).

To check the lower semicontinuity of \(\check{F}^{\mathcal{V}_{\Lambda}}\) recall from the Section 3 of [3] that

with \(Q (\mathbb{W}) = C (\| \mathbb{W} \|^p_{\mathfrak{S}} + 1)\) for some \(p \geqslant 1\). Furthermore observe that the function

is lower semicontinuous on \(\mathfrak{S} \times \mathfrak{H}_w\). Since the first marginal in \(\bar{\mathcal{X}}\) is fixed, the Portmanteau theorem gives, for any \(\mu_n \rightarrow \mu\) in \(\bar{\mathcal{X}}\):

Finally, by a slight adaptation of the proof of Lemma 21 below we see that there exists \(\delta > 0\) and \(C_{\delta, | \Lambda |} < \infty\) such that

for all \(\mu \in \bar{\mathcal{X}}\), from which the strong coercivity condition (8) follows.\(\Box\)

The admissibility of \(\check{F}^{\Lambda}\) implies the variational representation for \(\mathcal{W}_{\Lambda}\) and the existence of minimizers \(\mu^{f, \Lambda}\) of \(\check{F}^{f, \Lambda} := \check{F}^{f +\mathcal{V}_{\Lambda}}\) with the corresponding wFBSDE equation

Proof. As we already remarked the existence of \(\mu^{f, \Lambda}\) follows from the admissibility of \(\check{F}^{f, \Lambda}\). The functional \(Z \mapsto \mathcal{V}_{\Lambda} (\mathbb{W}, Z_{})\) is differentiable in \(H^1\) and

Provided \(Z \mapsto f (W + Z)\) is also differentiable, we have

which is (16).\(\Box\)

Remark 19. Note that \(Z \mapsto \mathcal{V}_{\Lambda} (\mathbb{W}, Z_{})\) is twice differentiable, with

Therefore if \(f\) is twice differentiable then we have also the second order condition:

for any minimizer \(\mu\) and any pair \(K, K'\) of admissible controls.

We record also the normalized variational representation for \(\mathcal{W}_{\Lambda} (f)\).

Proof. This follows from Lemma 13 by direct computation.\(\Box\)

Next we now show that the wFBSDE (16) gives us an a-priori bound uniform in \(| \Lambda |\) on the minimizer in terms of weighted norms of \(\mathbb{W}\). For this we first need some analytic bounds.

Proof. First observe that we have that \(\int_{\Lambda} \rho Z^4 = \| \mathbb{1}_{\Lambda} \rho^{1 / 4} Z^4 \|^4_{L^4}\), and \(\mathcal{E} (\rho^{1 / 2} Z) \geqslant C \| \rho^{1 / 2} Z \|^2_{H^1}\), for instance by fundamental theorem of calculus. By our assumptions on \(f\) we have, in the first case

While in the second case

For the second term, by Cauchy–Schwarz we obtain

For the third term, we use Lemma A.7 from [19] to get

For the final term we proceed similarly. By Lemma 16 we have \(\sup_{| \Lambda |} \| \mathbb{1}_{\Lambda} \|_{B_{p, p}^s (\rho^{\iota})} < \infty\) for any \(s < 1 / p\) and any \(\iota > 1 / 16\). Then with \(\varepsilon < 1 / 8\) and for any \(\iota > 1 / 16\) we have by fractional Leibniz rule

and we can choose \(\iota = 1 / 8 - \varepsilon > 1 / 16\) to get the claim. \(\Box\)

Proof. Note that the map \(Z \mapsto \rho Z\) is continuous from \(\mathfrak{H}\) to \(\mathfrak{H}\). Take \(K = \rho Z\) in (16) to get that any minimizer \(\mu^{f, \Lambda} \in \bar{\mathcal{X}}\) satisfies

From Lemma 23 and Lemma 21 below we deduce that the r.h.s. is controlled by a small fraction of the l.h.s. \(+\) a constant, from which we obtain the claim.\(\Box\)

Proof. We observe that \(m^2\) obviously commutes with \(\rho^{1 / 2}\), and

Using the assumptions (6) on the weight \(\rho\) we deduce

Let us now show that these estimates give tightness of \((\theta_{\Lambda})_{\Lambda}\) as \(| \Lambda | \rightarrow \infty\) on a suitable weighted Sobolev space.

Proof. Let \(g_N \in C^{\infty} (\mathbb{R}, R)\) be such that \(g_N (x) = x\) for \(| x | < N\) and \(g\). Define \(f^N (\phi) = g_N (\delta \| \phi \|^2_{H^{\varepsilon} (\rho^{1 / 2})})\) we have \(\nabla f^N (\phi) = (g_N)' (\| \phi \|^2_{H^{\varepsilon} (\rho^{1 / 2})}) \delta \langle D \rangle^{- 2 \varepsilon} \phi\) so

and this bound is uniform in \(N\). This implies by Lemma 21 that

and

And taking sup over \(N\) we get

a fact that implies exponential tightness of \((\theta_{\Lambda})_{\Lambda}\). For the second statement we proceed similarly. Indeed if \(\theta\) is an accumulation point of \((\theta_{\Lambda})_{\Lambda}\) then

This gives the claim provided that for \(f^N (\phi) = g^N (\delta \| \mathbb{1}_{\tilde{\Lambda}} \phi \|^4_{W^{- \varepsilon, 4} (\rho^{1 / 4})})\) we have

Indeed we compute \(\nabla f^N (\phi) = (g^N)' (\| \mathbb{1}_{\tilde{\Lambda}} \phi \|^4_{W^{- \varepsilon, 4} (\rho^{1 / 4})}) \delta \rho \mathbb{1}_{\Lambda} \langle \mathrm{D} \rangle^{- \varepsilon} (\langle \mathrm{D} \rangle^{- \varepsilon} \phi)^3\), from which we can check that

and \(\operatorname{supp} (\nabla f^N (\phi)) \subseteq \Lambda\). Applying Lemma 21 we get that the minimizer \(\mu^{f^N, \Lambda}\) satisfies

As a consequence, eq. (10) gives

which implies the claim.\(\Box\)

We have actually a more refined result: let \(\mu^{f, \Lambda}\) be a minimizer for \(\check{F}^{f, \Lambda}\). By Theorem 22 the family \((\mu^{f, \Lambda})_{\Lambda}\) is tight in \(\mathcal{P} (\mathfrak{S} \times \hat{\mathfrak{H}}_w)\).

Proof. Let the support of \(K\) be contained in \(B_N (0)\) for some \(N \geqslant 0\). Let \(\chi \in C^{\infty}_c (\mathbb{R}^2)\) such that \(\chi = 1\) on \(B_N (0)\) so that \(K = \chi K\). Then Lemma 21 implies that

where the constant depends on \(N\) but not on \(\Lambda\) and we have

Now observe that

where

Let \(\tilde{\chi} \in C^{\infty}_c (\mathbb{R}^2)\) and \(\tilde{\chi} = 1\) on the support of \(\chi\). It is not hard to see from the definition that

Indeed,

while, by using interpolation from the first to the second line, we have

and similarly

From this we get

The r.h.s. of this last equation is integrable by assumptions on \(K\) and bounds on \(Z\). From Lemmas 12 and 14 of [3] we know that there exist a family \((K^M)_{M \geqslant 1}\) adapted to the filtration generated by \((W_t)_t\) with

as \(M \rightarrow \infty\). After possibly passing to a subsequence, we can assume that \(\chi^{1 / 4} K^M \rightarrow \chi^{1 / 4} K\) in \(L_w^4 (\mu, L_w^4 (\mathbb{R}^2))\) where \(L_w^4\) denotes \(L^4\) with the weak topology, and in \(L_w^2 (\mu, L^2 (\mathbb{R}_+ \times \mathbb{R}^2))\). This together with the bound (20) implies that

At this point we have reduced the problem to showing that

since by assumption

This follows from the weak convergence \(\mu^{f, \Lambda} \rightarrow \mu\) and the bound

since we can approximate \(Z\) by bounded admissible controls \((Z^L)_L\) defined as \(Z^L_t = Z_{t \wedge T_L}\) with the stopping time

in such a way that

almost surely. Note that \(T_L \rightarrow 1\) as \(L \rightarrow \infty\) and therefore indeed \(Z^L_t \rightarrow Z_t\) as \(L \rightarrow \infty\).\(\Box\)

To summarize the result of this section: we obtained tightness of the finite volume \(\varphi^4_2\) measures and described any possible infinite volume limit via a novel kind of stochastic equation, the wFBSDE (19). Note that it cannot be interpreted anymore as the first order condition of a functional. Unfortunately, due to the strongly non-convex character of the renormalized potential we have currently limited information on its solutions. One expects that in the “single phase region” [17] (e.g. where \(m\) is large compared to \(\lambda\)) there should be uniqueness, and alternatively that multiple solutions should appear in “two phase region”. Despite these limitations eq. (19) is useful and in the next section we show that it allows to prove large deviations for \(\varphi^4_2\) in the semiclassical limit.

3Large deviations

In this section we want to investigate the large deviations of \(\varphi_2^4\) in the infinite volume in the semiclassical limit \(\hbar \rightarrow 0\) where \(\hbar > 0\) is a parameter which is inserted in the model as follows. For any \(f : \mathcal{S}' (\mathbb{R}^2) \rightarrow \mathbb{R}\), let \(\mathcal{W}^{\hbar}_{\Lambda} (f)\) be defined by

where

Here \(\theta^{\hbar}\) is the Gaussian measure on \(\mathcal{S}' (\mathbb{R}^2)\) with covariance \(\hbar (m^2 - \Delta)^{- 1}\) and the Wick ordering is now taken with respect to \(\theta^{\hbar}\). With some minor modifications to the derivation in the previous section we have the identity

where

Let \(\mu^{f, \Lambda, \hbar}\) a minimizer for this admissible \(\check{F}^{f, \Lambda, \hbar}\). Reasoning as in Lemma 24 above we have that the family \((\mu^{f, \Lambda, \hbar})_{\Lambda, \hbar}\) is exponentially tight for any bounded and continuous \(f\). For each \(\hbar > 0\) we pick a weakly convergent subsequence of \((\mu^{f, \Lambda, \hbar})_{\Lambda}\) with limit \(\mu^{f, \hbar}\). We have then also \(\theta^{\hbar}_{\Lambda} \rightarrow \theta^{\hbar} =\operatorname{Law}_{\mu^{f, \hbar}} (W_1 + Z_1)\) along the same subsequence and clearly, for all bounded continuous \(f\) we have

where \(\mathcal{W}^{\hbar} (f)\) is now defined as

In this setting where we do not have sufficient information to establish uniqueness of the limiting measures \((\mu^{f, \hbar})_{\hbar}\), nonetheless we are able to obtain a Laplace principle as \(\hbar \rightarrow 0\) for the corresponding family \((\theta^{\hbar})_{\hbar}\). Note that the family \((\theta^{\hbar})_{\hbar}\) is exponentially tight by Lemma 24.

Proof. Assume that \(f\) is of the form

for \(C \geqslant 0\) and \(\varphi \in C^{\infty}_c (\mathbb{R}^2)\).

It is enough to prove the theorem for this particular family of functions since we already know that \((\theta^{\hbar})_{\hbar}\) are exponentially tight, and the family given by (21) is rate function determining in the sense of Definition 3.15 of [16], as it isolates points and is bounded below.

From (12) we have

By taking a further subsequence we can also have that \(\mu^{\alpha f, \Lambda, \hbar} = \mu^{\alpha f, \Lambda, \hbar} \rightarrow \mu^{\alpha f, \hbar}\). We can then pass to the limit and obtain

for any \(f, \alpha\) and arbitrarily chosen accumulation point \(\mu^{\alpha f, \hbar}\). Note that we do not have to do a diagonal argument over the uncountably many choices of \(\alpha, f\): for any given \(\alpha, f\) we need just to take a subsequence of the sequence chosen to have \((\theta^{\hbar}_{\Lambda})_{\Lambda}\) converge in order to establish the equality.

Now recall that the minimizer \({\mu^{f, \hbar}} \) satisfies an infinite volume wFBSDE of the form

for any \(H\) with compact support. Furthermore we can carry over the proof of Lemma 21, and obtain that

This implies that the family \(\left( {\mu^{f, \hbar}} \right)_{\hbar}\) is also tight and therefore we can consider any accumulation point \(\mu^f\) as \(\hbar \rightarrow 0\) which will satisfy the same estimate. By passing to the limit in (22) via by now standard estimates similar to those in Lemma 25 we obtain

valid for any adapted and compactly supported \(H\). Note that this equation does not contain anymore the GFF. This equation has a unique solution. Indeed if \(\nu\) is another solution we can couple the two solutions in \(\rho \in \bar{\mathcal{Y}} (\mu^f)\) such that \(\nu =\operatorname{Law}_{\upsilon} (W, Z + K)\) and have

Now taking \(H = \chi K\) for \(\chi \in C^{\infty}_c (\mathbb{R}^2)\) and \(\chi \geqslant 0\), we obtain

By convexity of \(f\) and \((\cdot)^4\) we have that

Sending \(\chi \rightarrow 1\) we obtain that

Therefore \(\mu^f\) is the unique minimizer of the admissible (convex) functional

Let \(\nu^{\phi} =\operatorname{Law}_{\mathbb{P}} (W, s \mapsto s \phi)\) for \(\phi \in H^1\) then

Using the fact that the variational problem in the time variable for the functional

is minimized for \(\dot{Z}_s\) constant, we have for any \(\nu\)

As a consequence we obtain that

and that \(\mu^f = \nu^{\phi^f}\) for the unique \(\phi^f\) which attain the convex deterministic minimization problem on the r.h.s..

To sum up, we have established that

Integrating eq. (24) and using dominated convergence

This also implies that the derivative in \(\alpha \in [0, 1]\) of \(\alpha \mapsto \mathcal{W}^0 (\alpha f)\) exists and

To conclude it is enough to show that this coincides with

where

Recall that \(\phi^{\alpha f}\) is the minimizer of \(\phi \mapsto \alpha f (\phi) + J (\phi)\) so the Euler–Lagrange equation gives

At this point we note that

By convexity of \(f\) and strong convexity of \(J\) we have

Now

so

so dividing by \(\| \phi^{(\alpha + \gamma) f} - \phi^{\alpha f} \|_{H^1}\) we have

We can then calculate

Given that \(\)\(\| \phi^{ (\alpha + \gamma) f} - \phi^{\alpha f} \|_{H^1} \lesssim \gamma\) we have, since by Sobolev embedding \(\| \cdot \|^4_{L^4}\) is locally Lipschitz in the \(H^1\) norm

where in the last line we used the Euler–Lagrange equation for \(\phi^{\alpha f}\). This implies in turn

which concludes the proof.\(\Box\)

4Exponential interaction

In this section we will discuss a variational description of the exponential interaction, that is the measure

where now \(V^{\xi}\) is given by

with some \(\beta^2 \in (0, 8 \pi)\) and where \(M^{\beta} := \mathopen{\unicode{x27E6}} \exp (\beta \phi) \mathclose{\unicode{x27E7}}\) is the Gaussian Multiplicative Chaos (GMC) and \(\xi \in C^{\infty}_c (\mathbb{R}^2, \mathbb{R}_+)\) is a (smooth and compactly supported) spatial cutoff. We start with some preliminaries about the rigorous construction of the GMC.

Proof. For a proof of (25) see [7]. The property a) is due to Kahane [25], while the estimate in b) is due to Hoshino, Kawabi and Kusuoka [23].\(\Box\)

From these properties we can determine easily the Besov regularity of \(M^{\beta}\) following [23](Section 2).

Proof. We have

Recall that the Littlewood–Paley operators \((\Delta_i)_{i \geqslant - 1}\) are defined by \(\Delta_i = \psi (\mathrm{D} / 2^i)\) for \(i \geqslant 0\) and some Schwarz function \(\psi : \mathbb{R}^2 \rightarrow \mathbb{R}_{\geqslant 0}\). Now using translation invariance of the law of \(M^{\beta, \varepsilon}\) and the fact that \(\mathcal{F}^{- 1} \psi\) is a Schwarz function we estimate

This gives the statement, since by weak convergence of \(M^{\beta, \varepsilon} \rightarrow M^{\beta}\) we have

\(\Box\)

In this model we let \(\mathfrak{S} := \mathfrak{S}_0 \times \mathcal{P} (\mathbb{R}^2)\) and \(\mathbb{W}= (W, M^{\beta}) : \mathfrak{S}_0 \rightarrow \mathfrak{S}\) will be the appropriate enhanced GFF. By the usual abuse of notation we will denote also \(\mathbb{W}= (W, M^{\beta})\) the canonical process on \(\mathfrak{S}\).

We also need a slightly improved definition of the relaxed trial space \(\overline{\mathcal{X}}^{\exp}\).

For technical reasons we first establish the variational formulation for the approximate potential \(V^{\xi, \varepsilon}\) where

and the Wick ordering is now taken with respect to the covariance of \(W_{\varepsilon} = \psi_{\varepsilon} \ast W\).

Proof. Note that \(M^{\beta, \varepsilon}\) is positive and that Proposition 28 implies that for \(p\) close enough to \(1\) and \(\frac{1}{p} + \frac{1}{q} = 1\)

so \(V^{\xi}\) is a tame functional. From the properties of the exponential it is clear that

We have for all \(\mu \in \overline{\mathcal{X}}^{\exp}\):

from which also the coercivity of \(\check{F}^{V^{\xi, f, \varepsilon}}\) is clear. Furthermore it is not hard to see that

is continuous as a map \((\mathfrak{S}_0, H^1) \rightarrow \mathbb{R}\) (note hat here the renormalization if finite) and convergence of \(W, Z \text{ in } (\mathfrak{S}_0, H^1)\) implies convergence in \(L^{\infty} \times L^{\infty}\) of \(\psi_{\varepsilon} \ast Z, \psi_{\varepsilon} \ast W\). To conclude admissibility it remains to show that \(\check{F}^{V^{\xi, f, \varepsilon}}\) is lower semicontinuous on \(\overline{\mathcal{X}}^{\exp}\). Since we have already shown the interaction to be continuous it remains to do this for E. Indeed we have for any \(\mu \in \overline{\mathcal{X}}^{\exp}\) a sequence \((\mu_n)_n\) such that \(\mu_n \rightarrow \mu\) in \(\overline{\mathcal{X}}^{\exp}\)

this implies lower semicontinuity.\(\Box\)

We now show that we can remove the ultraviolet cutoff in the variational formulation. We introduce the unregularized functional

We will show that as \(\varepsilon \rightarrow 0\) \(\check{F}^{f, \xi, \varepsilon}\) \(\Gamma\)-converges to \(\check{F}^{f, \xi}\). We briefly recall the notion of \(\Gamma\)-convergence, see e.g [10] for a nice introduction.

Proof. We first show the liminf inequality. We have for sequence \((\mu_{\varepsilon})_{\varepsilon}\) such that \(\mu_{\varepsilon} \rightarrow \mu\) in \(\overline{\mathcal{X}}^{\exp}\)

Now let \(g^N : \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(g^N (x) = x\) if \(x < N\), \(g^N (x) = N\) if \(x > N\). Then \(g^N \leqslant N\), \(g^N\) is Lipschitz with constant \(1\) and \(g^N (x) \rightarrow x\). Set \(Z_t^N = g^N (Z_t)\) and \(\mu_{N, \varepsilon} =\operatorname{Law}_{\mu^{\varepsilon}} (W, Z^N)\). Then by monotone convergence

It remains to prove that

then the statement will follow from

We have that \(\| \exp (\beta Z^N) \|_{L^{\infty}} \leqslant \exp (\beta N)\) and by chain rule

By interpolation

provided \(2 / p' > \delta\). Now let \(p = p' / (p' - 1)\). Note that we can choose \(\delta\) such that \(\frac{8 \pi}{\beta^2} > p\) and \(\frac{\beta^2}{4 \pi} (p - 1) < \delta\) and \(2 / p' < \delta\) by taking

Since \(M^{\beta, \varepsilon} \rightarrow M^{\beta}\) in \(B_{p, p}^s\) by Proposition 28 this establishes that \(\int \xi \exp (\beta Z^{N, \varepsilon}) \mathrm{d} M^{\beta, \varepsilon} \rightarrow \int \xi \exp (\beta Z^N) \mathrm{d} M^{\beta}\) almost surely. Since

and the r.h.s is uniformly integrable in the probability variable, we have by dominated convergence

and we can conclude. To show the limsup inequality, for given \(\mu\) we need to construct \(\mu^{\varepsilon}\) such that

To do this we proceed similarly as in the first step. Define \(Z^{N, \delta} = g^N (\psi_{\delta} \ast Z)\), \(\mu^{N, \delta} =\operatorname{Law} (W, M^{\beta}, Z^{N, \delta})\), where \(\psi_{\delta}\) is a sequence of standard mollifiers. Then

weakly so

since \(Z^{N, \delta}\) is bounded and continuous. Furthermore \(\mathbb{E}_{\mu^{\varepsilon}} [\mathcal{E} (Z^{N, \delta})] \rightarrow \mathbb{E}_{\mu} [\mathcal{E} (Z^{N, \delta})]\). This implies that

Now by dominated/monotone convergence we have

and

so taking a suitable diagonal sequence implies the statement. \(\Box\)

Proof. The statement will follow from the fundamental theorem of \(\Gamma\)-convergence (see e.g. [14]), once we have established that \(\check{F}^{V^{\xi, \varepsilon}}\) is equicoercive on \(\overline{\mathcal{X}}^{\exp}\) , but this follows from

We can now relate this back to our general framework:

Proof. This follows mutatis mutandis as in Lemma 18.\(\Box\)

Proof. Observe that \(F^{0, \xi} (Z)\) is convex in \(Z\). Now let \(\mu, \nu\) be two minimizers for \(\check{F}^{0, \xi}\). Recall from Lemma 5 that there exists \(\upsilon \in \overline{\mathcal{Y}} (\mu)\) such that \(\operatorname{Law}_{\upsilon} (W, Z + K) = \nu\). Now

By Lemma 34 we have that

so

which implies \(K = 0\) and therefore \(\nu = \mu\). For the second statement observe that the mapping \(A : Z \rightarrow - | Z |\) leaves the energy \(\mathcal{E} (Z)\) invariant by chain rule, while

so if \(\mu^{0, \xi}\) is a minimizer, so is \(\operatorname{Law}_{\mu^{0, \xi}} (W, A (Z))\). By uniqueness we can conclude. For the third claim we test the wFBSDE with \(K = \rho Z\) and obtain

We can again estimate as in the proof of Lemma 21

and

Furthermore observe that

since \(\exp (\beta Z) Z\) is bounded below. Summing up, we have the estimate

where in the last bound we used Lemma 28.\(\Box\)

The proof of Lemma 36 is completely analogous to Lemma 20 and Lemma 18, thus we omit the details. Thanks to the convexity of the interaction we have a good control of the locality properties of the solutions to the wFBSDE (26).

Proof. Testing equation (26) with \(H = \rho^{- 2 N} K\) we have

Now

and we can compute, as in the proof of Lemma 21,

Furthermore observe that

Putting everything together we have

which implies the statement.\(\Box\)

Remark 39. Note that any sequence in \(\mathcal{Y}^M\) has a convergent subsequence.

Proof. This is a direct consequence of the decay of the minimizer established in Lemma 37.\(\Box\)

The next lemma discusses \(\Gamma\)-convergence of the variational description as the infrared cutoff is removed.

Proof. We first prove the liminf inequality. Assume that \(\upsilon^n \rightarrow \upsilon\) in \(\mathcal{Y}^M\). By definition of \(\xi_n\) we can restrict ourselves to \(\upsilon \in \mathcal{Y}^M (\mu)\). From the definition of convergence in \(\mathcal{Y}^M\) it is not hard to see that

and

by the Portmanteau theorem. Now we can invoke the Skhorohod's embedding theorem from [24] and pass to a probability space \(\tilde{\mathbb{P}}\) and on it a sequence \((W^n, M^{\beta, n}, Z^n, K)\) convergent almost surely on \(\mathfrak{S} \times \widehat{\mathfrak{H}}_w \times \mathfrak{H}_w (\rho^{- N})\) to \((W, M^{\beta}, Z, K)\) such that \(\operatorname{Law} (W^n, M^{\beta, n}, Z^n, K) = \upsilon^n\). Using the fact that

and almost-sure convergence we can assume that

for any \(\delta, p\) satisfying the assumptions of Proposition 28 and \(q < 2\). Let \(\varsigma_N : \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}_+\) be continuous such that

\(\varsigma_N (z, x) = 0 \text{if either } z \geqslant 2 N \text{or } | x | \geqslant 2 N\) and moreover \(\varsigma_N (z, y) \leqslant \varsigma_{N + 1} (z, y) .\) We will write in the following \(\varsigma_N (f, x)\) for the function \(x \in \mathbb{R}^2 \mapsto \varsigma_N (f (x), x)\). We claim that

Since

we can apply Lemma 45 below provided that, for any \(, q < 2\),

which in turn is ensured by Lemma 44 and Lemma 46. Then, recalling that \(Z \leqslant 0\) \(\mu^n\) almost surely and monotone convergence we have,

Let us now show that

To start, note that

so again we have that from Lemmas 44, 46 below, for any \(p < 2\),

From this and using that \(\xi_n = 1\) for \(| x | \leqslant n\), it is easy to see that

we have that,

To construct the recovery sequence consider \(\upsilon\) such that \(\bar{G}^{f, \infty, \exp} (\upsilon) < \infty\). Then

where the second inequality holds by definition of \(\mathcal{Y}^M\). Again, after an application of Lemma 44 and interpolation between \(L^{\infty}\) and \(H^1 (\rho^{- 1})\) and provided \(N > 1 / \delta\), we have that

So

Now set \(K^N (x) = \varsigma_N (K, x)\), by monotone convergence

Moreover, since \(| (\exp (\beta K^N) - 1)_- | \leqslant | (\exp (\beta K) - 1)_- |\), by dominated convergence also holds

Invoking [24] there exist a probability space \((\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{\mathbb{P}})\) and random variables \((W^n, M^{\beta}, Z^n)\) on it such that \((W^n, M^{\beta}, Z^n) \rightarrow (W, M^{\beta}, Z)\) almost surely in \(\mathfrak{S} \times \mathfrak{H}_w (\rho)\) and \(\operatorname{Law}_{\tilde{\mathbb{P}}} (W^n, M^{\beta}, Z^n) = \mu^n .\)

Let \(K\) be a random variable \(\tilde{\Omega} \rightarrow \mathfrak{H}_w (\rho^{- 1})\) such that \(\upsilon =\operatorname{Law}_{\tilde{\mathbb{P}}} (W, Z, K)\). Set again \(K^N = \varsigma^N (K)\) and \(\upsilon^{n, N} =\operatorname{Law}_{\tilde{\mathbb{P}}} (W^n, Z^n, K^N)\). We claim that

From Lemmas 44, 45, 46 we get

Furthermore

and \((K^N, Z^n) \rightarrow (K^N, Z)\) in \(H^1 (B (0, N))\). So again from Lemmas 44, 45, 46

Finally

follows by boundedness and continuity of \(f\). Taking a diagonal sequence we obtain the statement. \(\Box\)

Proof. This follows from \(\Gamma\)-convergence, the fact that \(\mathcal{Y}^M\) is sequentially compact and the fundamental theorem of \(\Gamma\)-convergence (see e.g. [14]).\(\Box\)

We conclude with some technical results needed in the proofs above.

Proof. Assume that \(g (0) = 0\). By definition of \(f\) we have

Furthermore by [15] p. 291–292, we have

Now for weak continuity, let \(\varphi_n \rightarrow \varphi\) weakly in \(W^{1, 2} (\rho^{\alpha})\). We know that

Now let \(\varphi_{n_k} \)be a subsequence such that \(f (\varphi_{n_k}) \rightarrow \xi\). We can extract a further subsequence (not relabeled) such that

almost surely. Then \(f (\varphi_{n_k}) \rightarrow f (\varphi)\) almost surely which in turn implies \(\xi = f (\varphi)\). From this we can conclude. The second statement follows from the first by writing \(g = \tilde{g} + g (0)\) with \(\tilde{g} (0) = 0\). \(\Box\)

Proof. By interpolation we have that for \(\delta < q / p'\) and \(N > \delta^{- 1}\) we have with

Now recall that for any \(\frac{8 \pi}{\beta^2} > p > 1\) and \(\delta > \frac{\beta^2}{4 \pi} (p - 1)\) we have

so provided \(1 / p + 1 / p' = 1\)

since \(p' \delta < q\). Note that for any \(\beta^2 < 8 \pi\) we can choose \(q < 2\) such that

so if we choose \(\delta\) such that

we obtain as required

Proof. It is enough to prove the statement for smooth functions by density. By Hölder's inequality we have

By the Leibniz rule

provided that \(\alpha N > 2\). By interpolation and provided that \(\alpha / p = 1 / 2\), we also have

Choosing \(N\) large enough gives the statement. For the second statement we rewrite

and proceed analogously.\(\Box\)

Appendix AOverview of the notation

Bibliography