Abstract
We present two approaches to establish the exponential decay of correlation functions of Euclidean quantum field theories (EQFTs) via stochastic quantization (SQ). In particular we consider the elliptic stochastic quantization of the Høegh–Krohn (or \(\exp (\alpha \phi )_2\)) EQFT in two dimensions. The first method is based on a path-wise coupling argument and PDE apriori estimates, while the second on estimates of the Malliavin derivative of the solution to the SQ equation.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The last decade has seen a renewed interest in the study of rigorous stochastic quantization (SQ) of Euclidean quantum field theories (EQFTs). SQ is a technique, first proposed by Nelson [43] and Parisi–Wu [48], to realize EQFTs, or more generally Gibbsian measures on \({\mathbb {R}}^d\) obtained as limits of perturbations of Gaussian measures, as solutions to certain stochastic partial differential equations (SPDEs) driven by Gaussian noise. After the pioneering work of Jona–Lasinio and Mitter [36, 37] and Da Prato–Debussche [16], only very recently substantial advances have allowed to attack the challenging problem of the SQ for classical EQFTs, including the \(\Phi ^4_3\) model, see e.g. [5, 15, 27, 28, 30, 31, 39, 41, 42].
While the original approach of Parisi–Wu to the SQ method based on a Langevin equilibrium diffusion gives rise to parabolic SPDEs, this it is not the only possibility. Nowadays we dispose of at least two other methods of stochastic quantization:
-
the elliptic SQ approach [1, 2, 14, 27], based on the dimensional reduction phenomenon described by Parisi and Sourlas [46, 47] and involving the solutions of an elliptic singular SPDE in \(d + 2\) dimensions;
-
the variational method [8, 12, 13] which involves forward–backward SDEs and can be also applied to fermionic EQFTs [17].
The aim of this work is to discuss the decay of correlations of Euclidean quantum fields from the point of view of the SQ methods. In particular we consider the elliptic SQ framework and restrict our attention to the following elliptic SQ equation with respect to the real valued random field \(\varphi (z), z \in {\mathbb {R}}^4\),
where \(\alpha \in {\mathbb {R}}\) and \(m > 0\). Here, \(\xi \) is a Gaussian white noise on \({\mathbb {R}}^4\) and \(- \infty \) means that the equation should be properly renormalized. The existence of a unique solution to Eq. (1) and the link with the corresponding EQF measure in two dimensions, called the Høegh–Krohn model [33] (also known as Liouville model in the literature) has been established in [2] for
More precisely, well-posedness holds in the weighted Besov space \(B_{p, p, \ell }^s ({\mathbb {R}}^4)\), for suitable (p, s) given in (5) and \(\ell > 0\) large enough (see Sect. 1.1 for precise notations).
The estimation of connected (or truncated) correlation functions, for example, the connected two-point function,
is a basic goal of any constructive EQFT approach. General truncated correlation functions allow to infer informations about masses of the particles in the QFT and estimate scattering amplitudes (see e.g. [32]). In the constructive literature, estimation of the connected correlation functions is obtained via cluster expansion methods or correlation inequalities. See for example the early work of Glimm–Jaffe–Spencer [25, 26]. The literature about expansion methods abounds. We suggest the interested reader to refer to [4, 6, 18, 24] and the reference therein for details and to [34] for a nice review of related results. Expansion methods for Euclidean fields involve two primary steps. The initial step is to expand the interaction into parts localized in different bounded volumes of Euclidean space. This gives control over the infinite volume method to establish the exponential decay of correlations. The second step is to expand interaction into components which are localized on different momentum scales. This helps in dealing with the local regularity properties of correlation functions. The technical difficulty is to mix these two expansions in a manageable way and to systematically extract contributions which require renormalization. Correlation inequalities methods instead employ discrete approximations, such as lattice approximations, whose specific algebraic properties allow for establishing bounds on a sufficiently broad class of observables.
While expansion methods can be applied to stochastic quantization, as evidenced in works such as [19, 38], we look here for a stochastic analytic approach leveraging the intrinsic features of SQ. Parisi [45] presented an early non-rigorous discussion of correlations within the SQ approach and studied how to estimate them directly via computer simulations. In this paper we introduce two simple, general and direct methods to study correlations in SQ applying them to the elliptic SQ of the exponential model (1):
-
Coupling approach
It is possible to infer the decay of truncated correlations by proving that the solutions to the SQ equation exhibit almost independent behaviour in different regions of space. This can be achieved by coupling the solution to two independent copies by suitably choosing the driving noises. As far as our knowledge extends, it has been Funaki [23] who first introduced this idea in the context of equilibrium dynamics of Ginzburg–Landau continuum models.
-
Malliavin calculus approach
Parisi [45] suggests to study variations of the SQ equations in order to infer truncated two-point correlations. His observation can actually be made precise and more general using the stochastic calculus of variations, i.e. the Malliavin calculus [44], and computing derivatives of the solutions to the SQ equation w.r.t. the driving noise \(\xi \).
These two approaches will be used to prove the following statement about a general class of truncated covariances:
Theorem 1
Let \(F_1, F_2\) be Lipschitz and functionals on \(B_{p, p, \ell }^s ({\mathbb {R}}^4)\) and f be a given smooth function supported in an open ball of unit radius around the origin. Then we have the following exponential decay
for all \(x_1, x_2 \in {\mathbb {R}}^4\) where the constant M depends on \(m, f, F_1, F_2\), the constant c depends on m but both are independent of \(x_1, x_2\).
Remark 1
Here \({\text {Cov}} (F, G) :={\mathbb {E}} [F G] -{\mathbb {E}} [F] {\mathbb {E}}[G]\) as usual and \((f \cdot \varphi (\cdot + x)) (\phi ):= \varphi (f (\cdot - x) \phi (\cdot ))\) for every test function \(\phi \).
In particular we prove that the solution of SQE (1) satisfies (formally),
It follows from Theorem 1 that the exponential EQFT in two dimensions has a mass gap, a fact first proven in [3] via correlation inequalities for the lattice approximation.
These approaches are general enough to be applicable to other EQFT models like \(P(\varphi )_2\) or \(\Phi ^4_3\) models. However a fundamental difficulty presents itself in establishing the required apriori estimates for the coupling method or in controlling the decay of Malliavin derivative in the Malliavin method. Both these difficulties originate in the lack of convexity of the renormalized interaction for a general EQFT. A similar problem is present in the analysis of logarithmic Sobolev inequalities for EQFT in bounded volumes [9, 11] especially for polynomial models. It also manifests in controlling the infinite volume limit of EQFT via stochastic quantization [13, 27, 28, 30], leading to a major obstacle in establishing uniqueness of the infinite volume solutions to the SQ equation.
Fortunately, these difficulties do not show up in the exponential model because its renormalization is multiplicative and it does not spoil the convex character of the interaction. For this reason our methods could be readily applied to obtain decay of correlations for the Sinh–Gordon model studied in [14]. Another model where uniqueness and correlations can be controlled via stochastic quantization is the Sine–Gordon model (for large mass and up the first renormalization threshold), studied by Barashkov via the variational method in [8]. Let us also mention that, inspired by the present paper, the coupling method has been already used to show decay of correlations for Euclidean fermionic QFTs and and for sine-Gordon Euclidean QFTs via the FBSDE SQ method, respectively in [17] and [29].
Let us stress that proving uniqueness of (any kind of) stochastic quantization and establishing decay of correlation of models like \(\Phi ^4_{2, 3}\) at high temperature is still largely an open problem which should be considered, in our opinion, as a crucial test to evaluate the effectivity of stochastic quantization as a constructive tool in quantum field theory. The present work is a preliminary step in the direction of understanding better this problem, and in general in devising appropriate tools to study stochastically quantized EQFTs.
Plan of the paper After introducing notations and definitions of function spaces in Sect. 1.1, the paper is structured into two main parts. In Sect. 2, we present a proof of Theorem 1 utilizing the coupling method, commencing with a review of essential results from [2]. Following this, in Sect. 3, we provide the Malliavin calculus proof of Theorem 1, beginning with a summary of relevant tools. The paper concludes with “Appendix A”, where we revisit a few necessary results from the literature and establish the existence and uniqueness of solutions to the approximate Eq. (35).
1.1 Notations
In this section we describe some notations and definitions of function spaces used across the whole paper. Some approach depending notations which are also used in the paper are discussed in the corresponding sections.
-
Throughout the paper, we use the notation \(a \lesssim b\) if there exists a constant \(c > 0\), independent of the variables under consideration, such that \(a \leqslant c b\). If we want to emphasize the dependence of c on the variable x, then we write \(a (x) \lesssim _x b (x)\). The symbol \(:=\) means that the right hand side of the equality defines the left hand side.
-
We set \({\mathcal {L}} :=- \Delta + m^2\).
-
For a distribution \(\varphi \), a smooth function f and \(x \in {\mathbb {R}}^d\), we define the translated distribution \((\varphi (\cdot + x))(\phi ) = \varphi (\phi (\cdot - x))\) for all test functions \(\phi \) and by \(f\cdot \varphi (\cdot + x)\) we denote the multiplication of a smooth function f and distribution \(\varphi (\cdot + x)\).
-
By \({\mathbb {N}}\) we understand the set of natural numbers \(\{ 1, 2, \ldots \}.\) For \(k \in {\mathbb {N}} \cup \{ 0 \}\), we write \(C^k ({\mathbb {R}}^d)\) to denote the set of real valued functions which are differentiable up to k-times and the k-th derivative is continuous. We write \(C ({\mathbb {R}}^d)\) for \(k = 0\) and the topology we consider on this space is uniform norm topology. By \(C_c^k ({\mathbb {R}}^4)\) we mean the collection of functions in \(C^k ({\mathbb {R}}^d)\) having compact support. We denote the the space of smooth functions having compact support by \(C_c^{\infty } ({\mathbb {R}}^4)\).
-
For any \(\ell > 0\) and weight \(r_{\ell } (x) :=(1 + | x |^2)^{- \ell / 2}\), by \(C_{\ell }^0 ({\mathbb {R}}^d)\) we denote the space of continuous functions on \({\mathbb {R}}^d\) such that
$$\begin{aligned} \Vert f \Vert _{C_{\ell }^0} {:=\sup _{x \in {\mathbb {R}}^d}} | f (x) r_{\ell } (x) | < \infty . \end{aligned}$$ -
By symbol \(L^p_{\ell } ({\mathbb {R}}^d), p \in [1, \infty ],\) we mean the Banach space of all (equivalence classes of) \({\mathbb {R}}\)-valued weighted p-integrable functions on \({\mathbb {R}}^d\). The norm in \(L^p_{\ell } ({\mathbb {R}}^d), 1 \leqslant p < \infty \) is given by
$$\begin{aligned} \Vert f \Vert _{L^p_{\ell }} :=\left[ \int _{{\mathbb {R}}^d} | f (y) r_{\ell } (y) |^p \textrm{d}y \right] ^{1 / p}, \qquad f \in L^p_{\ell } ({\mathbb {R}}^d). \end{aligned}$$For \(p = \infty \) we understand it with the usual modification. If \(\ell = 0\) we only write \(L^p ({\mathbb {R}}^d)\) instead \(L^p_0 ({\mathbb {R}}^d)\). Sometimes we also use weight function \(r_{\lambda , \ell } (x) :=(1 + \lambda | x |^2)^{- \ell / 2}\), for \(\lambda , \ell >0\), and in this case we define \(L^p_{\lambda , \ell } ({\mathbb {R}}^d)\) by writing \(r_{\lambda , \ell }\) in place of \(r_{\ell }\) in definition of \(L^p_{\ell } ({\mathbb {R}}^d)\). Similarly we define \(L^p_{\ell }(E)\) and \(L^p_{\lambda , \ell }(E)\) for an open subset \(E \subset {\mathbb {R}}^d\).
-
Let s be a real number and (p, q) be in \([1, \infty ]^2\). The weighted Besov space \(B_{p, q, \ell }^s ({\mathbb {R}}^d)\) consists of all tempered distributions \(f \in {\mathcal {S}}' ({\mathbb {R}}^d)\) such that the norm
$$\begin{aligned} \Vert f \Vert _{B_{p, q, \ell }^s} :=\left[ \sum _{j \geqslant - 1} 2^{s j q} \Vert \Delta _j (f) \Vert _{L^p_{\ell } ({\mathbb {R}}^d)}^q \right] ^{1 / q} \end{aligned}$$is finite, where \(\Delta _j\) are the non-homogeneous dyadic blocks. See Appendix A of [2] for details and properties of \(B_{p, q, \ell }^s ({\mathbb {R}}^d)\). We set \(C_{\ell }^2({\mathbb {R}}^d):= B_{\infty , \infty , \ell }^2 ({\mathbb {R}}^d)\).
-
For \(r > 0, x \in {\mathbb {R}}^d,\) we denote an open ball of radius r around x by B(x, r). We also use d(x, S) to define the distance between the point \(x \in {\mathbb {R}}^d \) and set \(S \subset {\mathbb {R}}^d.\)
-
Let \({\mathfrak {a}}\) be an auxiliary (radial) smooth, compactly supported function such that \({\text {supp}} {\mathfrak {a}} \subset B (0, 1)\), \(\int {\mathfrak {a}} (x) d x = 1,\) and \({\mathfrak {a}}_{\varepsilon } (x) :=\varepsilon ^{- 4} {\mathfrak {a}} (x / \varepsilon ), x \in {\mathbb {R}}^4\). Note that \({\text {supp}} {\mathfrak {a}}_{\varepsilon } \subset B (0, \varepsilon ).\)
Note that to save space we do not write the integration limit and the measure in the case when it is easily understood from the context.
2 The coupling approach
In this approach towards to proof of Theorem 1 we first prove (2) for a random field \(\varphi _{\varepsilon }\) which solves an approximation (7) of SPDE (1). Then due to Fatou’s lemma we pass to the limit \(\varepsilon \rightarrow 0\) and obtain (2) for \(\varphi \). We only need to consider the case of large \(l :=| x_1 - x_2 |\) in detail as for small l the estimate (2) holds trivially.
Let us now sketch briefly the idea of the coupling approach. We consider two open balls \(D_1\) and \(D_2\) in \({\mathbb {R}}^4\) of radius l/2 with centers \(x_1\) and \(x_2\). Further, we take two copies of Gaussian independent space white noises \(\zeta _1\) and \(\zeta _2\) and define, for \(i = 1, 2\),
In this way, in \(D_i\) we have that \(\xi = \xi _i\) for \(i = 1, 2\), while \(\xi _1\) and \(\xi _2\) are independent everywhere. We let \(X_{\varepsilon }, X_{1, \varepsilon }\) and \(X_{2, \varepsilon }\) be the solutions to linear part (cfr. (8)) of the approximations of the Eq. (1) with noises replaced by \(\xi _{\varepsilon }\), \(\xi _{1, \varepsilon }\) and \(\xi _{2, \varepsilon }\), respectively. Therefore \(X_{1, \varepsilon }\) and \(X_{2, \varepsilon }\) are independent while we will have \(X_{i, \varepsilon } \approx X_{\varepsilon }\) in \(D_i\). By stability estimates for eq. (1) we can derive estimates of the form (cfr. (23))
for some \({\mathfrak {c}}\) which depends on m and \({\mathfrak {p}}\), where \({\mathfrak {p}} \in [2, \infty )\) is fixed. In the above we have \(\varphi _{\varepsilon } = {\bar{\varphi }}_{\varepsilon } + X_{\varepsilon }\) and \(\varphi _{i, \varepsilon } = {\bar{\varphi }}_{i, \varepsilon } + X_{i, \varepsilon }\), where \({\varphi }_{\varepsilon }\) and \({\varphi }_{i, \varepsilon }\) respectively, are the unique solutions to the regularized SPDE (7) with \(\xi _{\varepsilon }\) and \(\xi _{i, \varepsilon }\) as detailed in Sect. 2.1. This estimate allows to replace \(\varphi _{\varepsilon }\) by \(\varphi _{i, \varepsilon }\) in \(D_i\) by paying a small error of the order \(e^{- c l}\) for some \(c>0\) (independent of \(x_i\)). Since \(\varphi _{1, \varepsilon }\) and \(\varphi _{2, \varepsilon }\) are independent, from the last estimate we can conclude easily the exponential decay for Lipschitz observables, see Sect. 2.2 for details.
2.1 Preliminaries
In this subsection we summarize the steps, with another suitably modified approximation, of the proof from [2], which also set further required notation. The main result of [2], which is about the existence of a unique solution to the singular SPDE (1), is based on the Da Prato–Debussche trick [16] and the fact that the Wick exponential is a positive measure.
-
Let us consider a complete probability space \((\Omega , {\mathfrak {F}}, {\mathbb {P}})\), which satisfies the usual hypothesis, and \(\xi \) as Gaussian white noise on \({\mathbb {R}}^4\) defined on \((\Omega , {\mathfrak {F}}, {\mathbb {P}})\).
-
Let X be the solution to \({\mathcal {L}}X = \xi \). The existence and uniqueness of such \(X \in B_{q, q, \ell }^{- \delta } ({\mathbb {R}}^4)\) for every \(q \in [1, \infty ], \delta > 0\) and \(\ell > 0\) is proved in [27].
-
To avoid clumsy notation we write \(\eta :=\exp ^{\diamond } (\alpha {\mathcal {L}}^{- 1} \xi )\) for the renormalized version of the distribution \(\exp (\alpha {\mathcal {L}}^{- 1} \xi - \infty )\), where \(\exp ^{\diamond }\) denotes the Wick exponential of the Gaussian distribution \(X ={\mathcal {L}}^{- 1} \xi \).
-
The first step in giving a meaning to Eq. (1) is to take the decomposition \(\varphi = {\bar{\varphi }} + X\). Then observe that formally \({\bar{\varphi }} \) satisfies
$$\begin{aligned} \begin{array}{l} {\mathcal {L}} {\bar{\varphi }} + \alpha \exp (\alpha {\bar{\varphi }}) \eta = 0. \end{array} \end{aligned}$$(3) -
For any \(\varepsilon > 0\) let us set \(\xi _{\varepsilon } :={\mathfrak {a}}_{\varepsilon } *\xi \) where \(*\) denotes convolution. Note that
$$\begin{aligned} \begin{array}{l} \eta = \sum _{k = 0}^{\infty } \frac{\alpha ^k}{k!} ({\mathcal {L}}^{- 1} \xi )^{\diamond k}, \end{array} \end{aligned}$$where \(\diamond \) denotes the Wick product and \(({\mathcal {L}}^{- 1} \xi )^{\diamond k}\)= \(\underbrace{{\mathcal {L}}^{- 1} \xi \diamond {\mathcal {L}}^{- 1} \xi \diamond \cdots \diamond {\mathcal {L}}^{- 1} \xi }_{k - {\text {times}}} = X^{\diamond k}\). By denoting \(X_{\varepsilon } ={\mathcal {L}}^{- 1} \xi _{\varepsilon }\) as the unique smooth solution to \({\mathcal {L}}X_{\varepsilon } = \xi _{\varepsilon }\), we set \(\eta _{\varepsilon }\) as the following positive measure
$$\begin{aligned} \eta _{\varepsilon } (\textrm{d}z) = \exp ^{\diamond } (\alpha {\mathcal {L}}^{- 1} \xi _{\varepsilon }) \textrm{d}z = \exp (\alpha {\mathcal {L}}^{- 1} \xi _{\varepsilon } - C_{\varepsilon }) \textrm{d}z, \end{aligned}$$(4)where \(C_{\varepsilon } :=\frac{\alpha ^2}{2} {\mathbb {E}} [| X_{\varepsilon } |^2]\). Moreover, from Section 3.1 of [2], we know that
$$\begin{aligned} \begin{array}{l} \eta _{\varepsilon } = \sum _{k = 0}^{\infty } \frac{\alpha ^k}{k!} ({\mathcal {L}}^{- 1} \xi _{\varepsilon })^{\diamond k}, \end{array} \end{aligned}$$and, for \(| \alpha | < 4 \sqrt{2} \pi , p \in (1, 2], s \leqslant - \frac{\alpha ^2 (p - 1)}{(4 \pi )^2}\) and \(\ell > 0\) large enough, \(\eta _{\varepsilon } \rightarrow \eta ,\) as \(\varepsilon \rightarrow 0\), in probability in \(B_{p, p, \ell }^s ({\mathbb {R}}^4).\) Note that the convergence \(\eta _{\varepsilon } \rightarrow \eta \) in probability implies that there exists a sequence \(\varepsilon _n\), which converges to 0, such that \(\eta _{\varepsilon _n} \rightarrow \eta ,\) as \(\varepsilon _n \rightarrow 0,\) in \(B_{p, p, \ell }^s ({\mathbb {R}}^4)\) \({\mathbb {P}}\)-almost surely. We will fix this sequence \(\{ \varepsilon _n \}_{n \geqslant 1}\) in the whole paper.
-
By Theorems 21 and 25 from [2] we have that for any \(| \alpha | < \alpha _{\max }\), there exist \(p, s, \delta \) satisfying
$$\begin{aligned} 1< p \leqslant 2, \quad p< \frac{2 (4 \pi )^2}{\alpha ^2}, \quad - 1< s \leqslant - \frac{\alpha ^2 (p - 1)}{(4 \pi )^2} \quad \text { and }\quad 0< \delta < s + 1,\nonumber \\ \end{aligned}$$(5)the Eq. (3) has a unique solution \({\bar{\varphi }}\) in \(B_{p, p, \ell + \delta '}^{s + 2 - \delta } ({\mathbb {R}}^4)\), \({\mathbb {P}}\)-almost surely, for large enough \(\ell > 0\) and small enough \(\delta ' > 0\). Moreover,
$$\begin{aligned} \begin{array}{l} \alpha {\bar{\varphi }} \leqslant 0 \end{array} \end{aligned}$$holds true. Furthermore, for \(\{ \varepsilon _n \}_{n \geqslant 1}\) as fixed above, \({\bar{\varphi }}_{\varepsilon _n} \rightarrow {\bar{\varphi }} \) in \(B_{p, p, \ell + \delta '}^{s + 2 - \delta } ({\mathbb {R}}^4)\) as \(n \rightarrow \infty \), \({\mathbb {P}}\)-almost surely, where \({\bar{\varphi }}_{\varepsilon _n}\) solves the approximate equation
$$\begin{aligned} {\mathcal {L}} {\bar{\varphi }}_{\varepsilon _n} + \alpha \exp (\alpha {\bar{\varphi }}_{\varepsilon _n}) \eta _{\varepsilon _n} = 0 \end{aligned}$$(6)uniquely in \(C_{\ell }^0 ({\mathbb {R}}^4)\) such that \(\alpha \varphi _{\varepsilon _n} \leqslant 0\).
-
Thus, for (p, s) such that (5) holds and \(\ell > 0\) large enough, \(\varphi = X + {\bar{\varphi }} \in B_{p, p, \ell }^s ({\mathbb {R}}^4),\) \({\mathbb {P}}\)-almost surely, solves SPDE (1) uniquely. If we consider the following approximation of SPDE (1)
$$\begin{aligned} \begin{array}{l} {\mathcal {L}} \varphi _{\varepsilon _n} + \alpha \exp (\alpha \varphi _{\varepsilon _n} - C_{\varepsilon _n}) ={\mathfrak {a}}_{\varepsilon _n} *\xi , \end{array} \end{aligned}$$(7)then, from the proof of Theorem 35 of [2], we know that \(\varphi _{\varepsilon _n} = {\bar{\varphi }}_{\varepsilon _n} + X_{\varepsilon _n}\) is the unique solution to (7) and \(\varphi _{\varepsilon _n} \rightarrow \varphi \) in \(B_{p, p, \ell }^s ({\mathbb {R}}^4)\), \({\mathbb {P}}\)-almost surely as \(n \rightarrow \infty \).
Let us recall that we have fixed the sequence of \(\{ \varepsilon _n \}_{n \in {\mathbb {N}}}\) which converges to 0 as \(n \rightarrow \infty \). To shorten the notation, we will write \(\varepsilon \rightarrow 0\) equivalently to \(n \rightarrow \infty \).
2.2 Proof of Theorem 1
Assume that \(| x_1 - x_2 | \leqslant 8.\) It is trivial to get (2) because its l.h.s. is bounded. Consider now the complementary case and let \(l :=| x_1 - x_2 | > 8\). Take two open balls \(D_1\) and \(D_2\) in \({\mathbb {R}}^4\) of radius l/2 with centers \(x_1\) and \(x_2,\) respectively. Further, we take two copies of Gaussian independent space white noises \(\zeta _1\) and \(\zeta _2\) defined on \((\Omega , {\mathfrak {F}}, {\mathbb {P}})\). Define the processes \(X_1\) and \(X_2\) as follows:
Note that that \(\xi - \xi _i = 0\) on \(D_i, i = 1, 2\) in the sense of distributions \({\mathbb {P}}\)-a.s. Moreover, since \(D_1 \cap D_2 = \emptyset \), the processes \(X_1\) and \(X_2\) are independent. Indeed, by setting \((\mathbb {1}_{D_i} \xi ) (f) :=\xi (\mathbb {1}_{D_i} f)\), we observe that for \(f, g \in L^2 ({\mathbb {R}}^4),\)
Let us set \(\xi _{\varepsilon } :={\mathfrak {a}}_{\varepsilon } *\xi \) and \(\xi _{i, \varepsilon } :={\mathfrak {a}}_{\varepsilon } *\xi _i, i = 1, 2\) for the whole subsection. Let \(\varphi _{\varepsilon }\) and \(\varphi _{i, \varepsilon }\), respectively, be the unique solutions to the following regularized version of eq. (1)
and
where \(C_{\varepsilon } :=\frac{\alpha ^2}{2} {\mathbb {E}} [| X_{\varepsilon } |^2] = \frac{\alpha ^2}{2} {\mathbb {E}} [| X_{i, \varepsilon } |^2] \) for \(X_{\varepsilon } ={\mathcal {L}}^{- 1} \xi _{\varepsilon }\) and \(X_{i, \varepsilon } ={\mathcal {L}}^{- 1} \xi _{i, \varepsilon }, i = 1, 2\). Note that due to stationarity in space of the white noise \(\xi \), the constant \(C_{\varepsilon }\) does not depend on \(x \in {\mathbb {R}}^4\).
Next, let us fix \({\mathfrak {p}} \in [2, \infty )\) and consider \(\varphi \), \(\varphi _1\) and \(\varphi _2\) as the unique solutions to the SPDE (1) with noises \(\xi \), \(\xi _1\) and \(\xi _2\), respectively. Their existence has been summarized in Sect. 2.1. Then observe that, since \(F_1\) and \(F_2\) are Lipschitz and bounded functionals, using the Hölder inequality we get the following
Here we used that, since the processes \(\xi _1\) and \(\xi _2\) are independent and the processes \(\xi ,\xi _1\) and \(\xi _2\) have same law,
But thanks to Fatou’s lemma (see Theorem 2.72 of [10]), to get (2) from (9) it is enough to prove that, for \(i = 1, 2\),
uniform in \(\varepsilon \), for some \(c > 0\) which does not depend on \(x_1, x_2\).
Due to symmetry, it is sufficient to estimate \({\mathbb {E}} [\Vert f \cdot \varphi _{\varepsilon } (\cdot + x_1) - f \cdot \varphi _{1, \varepsilon } (\cdot + x_1) \Vert _{B_{p, p, \ell }^s}^{{\mathfrak {p}}}]\). For that let \({\tilde{D}}_1 :=B \left( x_1, \frac{l}{4} \right) \) and take
a weight function where we set the value of \(\beta \) later. Further, let us take \(\theta \) as a non-negative smooth function supported in \({\tilde{D}}_1\) such that \(\theta = 1\) in \({\bar{D}}_1 :=B \left( x_1, \frac{l}{8} \right) \). To shorten the notation we also set \({\bar{\rho }} (x):= \theta (x) \rho (x)\).
Since f has support in B(0, 1), by the Besov embedding Theorem 5 followed by continuous embedding of \(L_{\ell }^{{\mathfrak {p}}}({\mathbb {R}}^4)\) into \(B^0_{{\mathfrak {p}}, \infty , \ell }({\mathbb {R}}^4)\) we get, where \(\chi _{\varepsilon } :=\varphi _{\varepsilon } - \varphi _{1, \varepsilon }\),
Towards estimating \(\Vert {\bar{\rho }} \chi _{\varepsilon } \Vert ^{{\mathfrak {p}}}_{L^{{\mathfrak {p}}} (B (x_1, 1))} \), first we claim that
This is obvious for \(x \in {\mathbb {R}}^4 \setminus {\tilde{D}}_1\). So let us take \(x \in {\tilde{D}}_1\). Since
it is sufficient to show that \((\xi - \xi _1, {\mathfrak {a}}_{\varepsilon } *g)_{{\mathcal {S}}', {\mathcal {S}}} = 0\) for all \(g \in C_c^{\infty } ({\tilde{D}}_1)\), where \((\cdot , \cdot )_{{\mathcal {S}}', {\mathcal {S}}}\) is duality between Schwartz function \({\mathcal {S}}\) and Schwartz distribution \({\mathcal {S}}'\). But, since \(\xi - \xi _1 = 0\) on \(D_1\), for this it is enough to show that \({\text {supp}} ({\mathfrak {a}}_{\varepsilon } *g) \subset D_1\). This follows because
and for \(z \in D_1^c\) and \(y \in {\tilde{D}}_1\), \(| z - y | \geqslant \frac{l}{4} > \frac{\varepsilon l}{4}\). Hence the claim (12).
Next, observe that \(\chi _{\varepsilon }\) satisfies
where \(Q_{\varepsilon } :=\alpha ^2 \int _0^1 \exp \{ \alpha \varphi _{1, \varepsilon } - C_{\varepsilon } + \Theta \alpha (\varphi _{\varepsilon } - \varphi _{1, \varepsilon }) \} \textrm{d}\Theta > 0\). Then, testing (13) with \({\bar{\rho }}^{{\mathfrak {p}}} | \chi _{\varepsilon } |^{{\mathfrak {p}}- 2} \chi _{\varepsilon }\) and integrating on \({\mathbb {R}}^4\) give
where the noise term vanishes because of (12). The first term on the l.h.s. above can be expanded as
But, since the integration by parts and the definition of divergence give
we have
Thus, substitution of (15) into (14) together with \(Q_{\varepsilon } > 0\) yield
Furthermore, since the integration by parts and the product rule of derivative give
from inequality (16) we obtain
Since \(\nabla \rho (x) = - m \beta \frac{x - x_1}{| x - x_1 |} \rho (x)\) for \(x \in {\mathbb {R}}^4 \setminus \{ x_1 \}\),
inequality (18) yield
where to get the r.h.s. terms we also used \((a+b)^2 \le 2(a^2 + b^2)\), \(\forall a,b \in {\mathbb {R}}\). Consequently, by regrou** the terms together with \(\bigg \vert \frac{x - x_1}{| x - x_1 |} \cdot \nabla \theta \bigg \vert \le |\nabla \theta |\) we get
Moreover, since \(\theta \) is supported in \({\tilde{D}}_1\) and \(\theta = 1\) on \({\bar{D}}_1\), (19) gives
To keep the coefficient of \(\Vert {\bar{\rho }} \chi _{\varepsilon } \Vert _{L^{{\mathfrak {p}}}}\) positive in the l.h.s. above, we choose \(\beta = \beta ({\mathfrak {p}}) > 0\) so small such that
To keep the notation simpler we set
Thus, from (20) we deduce that
where \(M_{\theta } > 0\) is the bound of \(\theta \) and its derivatives up to order 2.
Further, since
by substituting (22) in (11) we infer that
Thus, by applying \({\mathbb {E}}\) on both sides we get
To estimate the term \({\mathbb {E}} [\Vert X_{\varepsilon } - X_{1, \varepsilon } \Vert _{L^{{\mathfrak {p}}} ({\tilde{D}}_1 {\setminus } {\bar{D}}_1)}^{{\mathfrak {p}}}]\), since \({\text {supp}} {\mathfrak {a}}_{\varepsilon } \subset B (0, \varepsilon ),\) we first infer that \(\xi _{\varepsilon } = \xi _{1, \varepsilon }\) on \(D_{1, \varepsilon } :=B \left( x_1, \frac{l}{2} - \varepsilon \right) \). By using the representation from Lemma 6 we have that
Since, for \(x \in {\tilde{D}}_1 \setminus {\bar{D}}_1\), we have \(| x - z | > \frac{l}{4} - \varepsilon \gg 1\) for \(z \in D_{1, \varepsilon }^c\), thus by Lemma 6 (1) we obtain
which is finite and independent of \(\varepsilon \). Here we have also employed the fact that \(\int _{{\mathbb {R}}^4} ({\mathfrak {a}}_{\varepsilon } *{\mathfrak {a}}_{\varepsilon }) (z - z_1) \textrm{d}z = 1\), which holds true because \({\mathfrak {a}}_{\varepsilon } *{\mathfrak {a}}_{\varepsilon }\) approximates \(\delta *\delta \), where \(\delta \) represents the Dirac delta distribution.
Consequently, since \((X_{\varepsilon } - X_{1, \varepsilon }) (x)\) is Gaussian from (24), by hypercontractivity (see Theorem 3.50 in [35]) there exists a constant \(C_{{\mathfrak {p}}} > 0\) such that, for every \(x \in {\tilde{D}}_1 \setminus {\bar{D}}_1\),
Furthermore, since
the Fubini Theorem followed by (25) yield
Finally, we assert that \({\mathbb {E}} [\Vert {\bar{\varphi }}_{\varepsilon } \Vert _{L^{{\mathfrak {p}}}({\mathbb {R}}^4)}^{{\mathfrak {p}}}] < \infty \). This assertion trivially implies \({\mathbb {E}} [\Vert {\bar{\varphi }}_{\varepsilon } \Vert _{L^{{\mathfrak {p}}} ({\tilde{D}}_1 {\setminus } {\bar{D}}_1)}^{{\mathfrak {p}}}] < \infty \) in (23). We start the proof of this claim by recalling from Sect. 2.1 that \(\alpha {\bar{\varphi }}_{\varepsilon } \leqslant 0\) and \({\bar{\varphi }}_{\varepsilon }\) is a unique solution to
By testing (27) with \(\rho ^{{\mathfrak {p}}} | {\bar{\varphi }}_{\varepsilon } |^{{\mathfrak {p}}- 2} {\bar{\varphi }}_{\varepsilon }\) and integrating it on \({\mathbb {R}}^4\) we obtain
where \(\int \rho ^{{\mathfrak {p}}- 1} | {\bar{\varphi }}_{\varepsilon } |^{{\mathfrak {p}}- 2} {\bar{\varphi }}_{\varepsilon } (- \Delta (\rho {\bar{\varphi }}_{\varepsilon })) \ge 0\), (28) gives
Since \(\eta _{\varepsilon } \rho ^{{\mathfrak {p}}}\) is a positive distribution, the second l.h.s. term in (29) can be estimated as
where \({\mathbb {I}}: {\mathbb {R}} \rightarrow {\mathbb {R}}_+\) is a smooth function supported on \((- \infty , 1)\). Note that \({\mathbb {I}} (\alpha {\bar{\varphi }}_{\varepsilon }) = 1\), since \(\alpha {\bar{\varphi }}_{\varepsilon } \leqslant 0\). But, for each \(x \in {\mathbb {R}}^4\),
for some \(C > 0\), where the r.h.s is independent of \(\varepsilon \) and x. By substituting the above estimate into (29) we obtain
Now since \(\nabla \rho (x) = - m \beta \frac{x - x_1}{| x - x_1 |} \rho (x)\) for \(x \in {\mathbb {R}}^4 {\setminus } \{ x_1 \}\) and \(\Delta \rho = \rho m^2 \beta ^2\), we can choose \(\beta > 0\) such that
Consequently, with \(\beta \) such that (21) and (31) hold true, from (30) we deduce that
Thus, \({\mathbb {E}} \left[ \Vert \rho {\bar{\varphi }}_{\varepsilon } \Vert _{L^{{\mathfrak {p}}}}^{{\mathfrak {p}}} \right] < \infty \) and the bound is uniform in \(\varepsilon \) because
Similarly we can show that \({\mathbb {E}} [\Vert \rho {\bar{\varphi }}_{1, \varepsilon } \Vert _{L^{{\mathfrak {p}}}}^{{\mathfrak {p}}}] < \infty \) uniformly in \(\varepsilon \).
Hence, substituting (26) together with (32) and the uniform boundedness of \({\mathbb {E}} [\Vert \rho {\bar{\varphi }}_{\varepsilon } \Vert _{L^{{\mathfrak {p}}}}^{{\mathfrak {p}}}]\) and \({\mathbb {E}} [\Vert \rho {\bar{\varphi }}_{1, \varepsilon } \Vert _{L^{{\mathfrak {p}}}}^{{\mathfrak {p}}}]\) from (23), for \(\beta \) satisfying (21) and (31), we have
which is independent of \(\varepsilon \) and \(x_1\). Here we have also used \(e^{m \beta \left( 1 - \frac{l}{8} \right) } \simeq _{m, {\mathfrak {p}}} e^{- m \beta \frac{l}{8}}\). Hence we get (10) and due to inequality (9) the proof of Theorem 1 is complete.
3 The Malliavin calculus approach
In this section our aim is to present the proof of Theorem 1 via the approach based on Malliavin calculus. The proof will start by considering an approximation \(\varphi _{\varepsilon , R}\) useful to be able to apply easily the Malliavin calculus, see eqns. (35) and (36). The solution theory to (35) is closely related to Lemmata 30 and 31 of [2] and proved in Proposition 1 and Lemma 5 below. The Malliavin calculus enters in estimating \({\text {Cov}} (\varphi _{\varepsilon , R} (x_1), \varphi _{\varepsilon , R} (x_2))\) in terms of the Malliavin derivative of \(\varphi _{\varepsilon , R}\) which we denote by \(D_z \varphi _{\varepsilon , R}\), see eqs. (60), (62) and (63). The existence of \(D_z \varphi _{\varepsilon , R}\) and the linear elliptic SPDE it satisfies are established in Theorem 3 thanks to a preliminary abstract result from [49] which we state as Theorem 2. Finally the Feynman–Kac formula and some estimates from Malliavin calculus, for example (61), help us to finish the proof.
3.1 Preliminaries
Before moving on, let us first recall the tools from Malliavin calculus that we will need. Most of the definitions and preliminary results here are taken from Chapter 1 of Nualart’s book [44]. Let H be a separable Hilbert space and \(W = \{ W (h), h \in H \}\) an isonormal Gaussian process defined on a complete probability space \((\Omega , {\mathfrak {F}}, {\mathbb {P}})\). Let \({\mathcal {E}}\) be the \(\sigma \)-field generated by the random variables \(\{ W (h), h \in H \}\). Since \({\mathcal {E}} \subseteq {\mathfrak {F}}\), note that when we write \((\Omega , {\mathcal {E}}, {\mathbb {P}})\) we mean that \({\mathbb {P}}\) is the restriction of the probability measure defined on \({\mathfrak {F}}\) to \({\mathcal {E}}\).
For each \(n \geqslant 0\) by \(H_n (x)\) we denote the well known nth Hermite polynomial and by \({\mathcal {H}}_n,\) the Wiener chaos of order n, that is, the closed linear subspace of \(L^2 (\Omega , {\mathcal {E}}, {\mathbb {P}})\) generated by the random variables \(\{ H_n (W (h)), h \in H, \Vert h \Vert _{H} = 1 \}\) whenever \(n \geqslant 1\), and the set of constants for \(n = 0\). One of the important results in the Malliavin calculus is the Wiener chaos decomposition of \(L^2 (\Omega , {\mathcal {E}}, {\mathbb {P}})\) into its projections in the spaces \({\mathcal {H}}_n\), i.e.,
In particular for any \(F \in L^2 (\Omega , {\mathcal {E}}, {\mathbb {P}})\), we have \(F = \sum _{n = 0}^{\infty } J_n F\) where \(J_n F\) denotes the projection of F into \({\mathcal {H}}_n\). We will restrict our discussion of this section to \(L^2 (\Omega , {\mathcal {E}}, {\mathbb {P}})\) and to shorten the notation we will denote it by \(L^2 (\Omega )\).
The Malliavin derivative operator D maps the domain \({\mathbb {D}}^{1, 2} \subseteq L^2 (\Omega ) \) to the space of H-valued random variables \(L^2 (\Omega ; H).\) Note that \(F \in {\mathbb {D}}^{1, 2}\) if and only if \(\sum _{n = 1}^{\infty } n \Vert J_n F \Vert _{L^2 (\Omega )}^2 < \infty \). Moreover, in this setting for all \(n \geqslant 1\), we have
The divergence operator \(\delta : {\text {Dom}} \delta \subseteq L^2 (\Omega ; H) \rightarrow L^2 (\Omega )\) is defined as the adjoint of the derivative operator D. We will work in the special case of \(H = L^2 (T, {\mathcal {B}}, \tau )\), where \((T, {\mathcal {B}})\) is a measurable space and \(\tau \) is a \(\sigma \)-finite atom-less measure on \((T, {\mathcal {B}})\). Also, we will identify \(L^2 (\Omega ; L^2 (T))\) with \(L^2 (T \times \Omega )\) which is the set of square integrable stochastic processes. Thus, for \(F \in {\mathbb {D}}^{1, 2}\), \(D F \in L^2 (T \times \Omega )\) and we write \(D_t F = D F (t), ~ \forall t \in T\). By \({\mathbb {D}}^{1, 2} (L^2 (T))\) we denote the set of stochastic processes \(u \in L^2 (T \times \Omega )\) such that \(u (t) \in {\mathbb {D}}^{1, 2}\) for almost all \(t \in T\) and there exists a measurable version of the two parameter process \(\{ D_s u (t) \}_{s, t \in T} \subset L^2 (\Omega )\) satisfying
In the Malliavin calculus literature, the space \({\mathbb {D}}^{1, 2} (L^2 (T))\) is generally denoted by \({\mathbb {L}}^{1, 2}\). Note that \({\mathbb {L}}^{1, 2}\) is a subset of \({\text {Dom}} \delta \) and isomorphic to \(L^2 (T; {\mathbb {D}}^{1, 2}).\) Then, see (1.54) of [44], for \(u, v \in {\mathbb {L}}^{1, 2}\) we have
Let \(\{ P_t, t \geqslant 0 \}\) be the one parameter Ornstein-Uhlenbeck semigroup of contraction operators in \(L^2 (\Omega )\) and by \(L: L^2 (\Omega ) \ni F \rightarrow \sum _{n = 0}^{\infty } - n J_n F \in L^2 (\Omega )\) we denotes its infinitesimal generator with domain
From Proposition 1.4.3 of [44] we know that, for \(F \in L^2 (\Omega ),\) \(F \in {\text {Dom}} L\) if and only if \(F \in {\mathbb {D}}^{1, 2}\) and \(D F \in {\text {Dom}} \delta \). In this case we have \(\delta D F = - L F.\)
With the above notation, equality (90) in [21] gives the following commutation property
and the proof of Lemma B.1 in [21] give the following first order expansion
To proceed with our analysis, let us fix the \(\sigma \)-finite measure space \((T, {\mathcal {B}}, \tau )\) as \(({\mathbb {R}}^4, {\mathcal {B}} ({\mathbb {R}}^4), \textrm{d}x)\) where \({\mathcal {B}} ({\mathbb {R}}^4)\) denotes the Borel \(\sigma \)-field on \({\mathbb {R}}^4\) and \(\textrm{d}x\) stands for the Lebesgue measure.
3.2 Proof of Theorem 1
We recall that \(\xi \) is a given space white noise on \({\mathbb {R}}^4\). Thus, the isonormal Gaussian process we consider here is \(W (h) = \langle \xi , h \rangle , h \in L^2_{\ell } ({\mathbb {R}}^4)\), indexed by the Hilbert space \(L^2_{\ell } ({\mathbb {R}}^4).\) We will be working under the framework of Malliavin calculus associated to white noise \(\xi \) on \({\mathbb {R}}^4\). To setup, let \(\Omega = B^{- 2 - \kappa }_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) and let \({\mathbb {P}}\) be the law of \(\xi \) on \(\Omega .\)
It turns out that the following approximation of the Eq. (3), instead of (6), is more suitable to work with the above mentioned tools from Malliavin calculus
where \(K_R: (0, \infty ) \rightarrow (0, \infty )\) is a smooth function which is equal to x if \(x \in (0, R - 1]\), equal to R if \(x \geqslant R\) and \(K_R\) is increasing for \(x \in (R - 1, R)\). Since the proof presented here of the solution theory to Eq. (35) is closely related to Lemmata 30 and 31 of [2], the results about the existence of a unique solution \({\bar{\varphi }}_{\varepsilon , R}\) to (35) are postponed to Proposition 1 and Lemma 5 in Appendix A. Moreover, it is straightforward to see that \({\bar{\varphi }}_{\varepsilon , R} \rightarrow {\bar{\varphi }}_{\varepsilon }\) as \(R \rightarrow \infty \), where \({\bar{\varphi }}_{\varepsilon }\) is the unique solution to the Eq. (6).
Further recall, from (4), that we denote the expression \(\exp (\alpha X_{\varepsilon } - C_{\varepsilon })\) by \(\eta _{\varepsilon }\). Let us define the following random field
where \({\mathcal {G}}\) is the Green function associated with the operator \((- \Delta + m^2)^{- 1}\) and \({\mathcal {G}}_{\varepsilon } :={\mathfrak {a}}_{\varepsilon } *{\mathcal {G}}\). It can be shown that \({\mathcal {G}}_{\varepsilon } *\xi \) is a smooth Gaussian process, see Theorem 5.1 of [41].
By setting \(\varphi _{\varepsilon , R} = {\bar{\varphi }}_{\varepsilon , R} + X_{\varepsilon }\), from (35) we get that \(\varphi _{\varepsilon , R}\) uniquely solves the following equation
which is equivalent to say that, for \(x \in {\mathbb {R}}^4\) and \(\omega \in \Omega \),
To shorten the notation we will write
Since one can write the term \({\text {Cov}} (F_1 (f \cdot \varphi _{\varepsilon , R} (\cdot + x_1)), F_2 (f \cdot \varphi _{\varepsilon , R} (\cdot + x_2)))\), that we want to estimate, in terms of \(D_z \varphi _{\varepsilon , R}\), see (60) for precise expression, we aim next to find the equation for \(D_z \varphi _{\varepsilon , R}\). This we achieve in Theorem 3 whose proof is based on the following abstract result which is stated as Theorem 2.5 in [49].
Theorem 2
Let \((\Omega , {\mathbb {P}})\) be a complete probability space on which \(\xi \) is a canonical process. Further, assume that H is continuously embedded in \(\Omega \) and let us denote this embedding by i. Let \(F \in L^2 (\Omega )\). Then \(F \in {\mathbb {D}}^{1, 2}\) iff the following conditions are satisfied.
-
1.
For all \(h \in H\), there exists a version \({\tilde{F}}_h\) of F such that, for every \(\omega \in \Omega \), the map** \({\mathbb {R}} \ni t \mapsto {\tilde{F}}_h [\omega + t i (h)]\) is absolutely continuous.
-
2.
There exists \(\varsigma \in L^2 (\Omega ; H)\) such that, for all \(h \in H,\)
$$\begin{aligned} \lim _{t \rightarrow 0} \frac{1}{t} \{ F [\omega + t i (h)] - F (\omega ) \} = \langle \varsigma (\omega ), h \rangle , \qquad {\mathbb {P}} \text {-a.s.} \end{aligned}$$
From the proof of Theorem 3 it can be observed that we apply Theorem 2, for each \(x \in {\mathbb {R}}^4,\) \(\varepsilon \) and R on F with \(H :=L_{\ell }^2 ({\mathbb {R}}^4)\) where
Since most of the results of this section are independent of \(\varepsilon , R\) and x or for fixed \(\varepsilon , R\) and x, unless otherwise stated we will not write the explicit dependence of functions defined here on \(\varepsilon , R\) and x.
To study the required properties of F, which allow us to apply Theorem 2, we write (37) in the functional form as, for \(\omega \in \Omega \),
Here \({\mathcal {T}}\) is defined as
Note that, because of the convolution, the map \({\mathcal {T}}\) is well-defined. Moreover, by definition of the map \({\mathcal {T}}\), (38) can be understood as, for each \(\omega \in \Omega \),
Thus, because of (40), in order to study F we first show in Lemma 2 that \({\mathcal {T}}^{- 1}\) exists, i.e., prove the bijectivity of the map \({\mathcal {T}}\). This is precisely our next result. Before this we prove an auxiliary result as follows.
Lemma 1
Let \({\mathfrak {v}} \in L_{\ell }^2 ({\mathbb {R}}^4) \) and \({\mathfrak {u}} \in H_{\ell }^2 ({\mathbb {R}}^4)\) be a unique weak solution to \((- \Delta + m^2) {\mathfrak {u}}={\mathfrak {v}}\). Then
where \(\langle \cdot , \cdot \rangle \) and \(\langle \cdot , \cdot \rangle _{\ell }\), respectively, denote the standard inner product in \(L^2 ({\mathbb {R}}^4)\) and \(L^2_{\ell } ({\mathbb {R}}^4)\).
Proof
Let \({\mathfrak {u}}={\mathcal {G}} *{\mathfrak {v}} \in H_{\ell }^2 ({\mathbb {R}}^4)\) be a unique weak solution to \((- \Delta + m^2) {\mathfrak {u}}={\mathfrak {v}}\) for given \({\mathfrak {v}} \in L_{\ell }^2 ({\mathbb {R}}^4)\).
Multiplying on both sides of \((- \Delta + m^2) {\mathfrak {u}}={\mathfrak {v}}\) by \(r_{\ell }^2 {\mathfrak {u}}\) give
Integration by parts yield,
By substituting \({\mathfrak {u}}={\mathcal {G}} *{\mathfrak {v}}\), above gives the conclusion. \(\square \)
To avoid complexity in notation we set \(G (w) :=\alpha K_R (\exp (\alpha w - C_{\varepsilon })), w \in {\mathcal {B}}\). Then, G is non-negative, bounded, smooth and non-decreasing.
Lemma 2
The map \({\mathcal {T}}\) is bijective from \({\mathcal {B}}\) onto \({\mathcal {B}}\).
Proof
Let us first show that \({\mathcal {T}}\) is one-one. In particular, we show that for small enough \(\lambda > 0\) if \(u, v \in {\mathcal {B}} \subset L_{\lambda , \ell '}^2 ({\mathbb {R}}^4)\) for \(\ell \leqslant \ell '\) such that \({\mathcal {T}}u ={\mathcal {T}}v\), then \(u = v\).
Since \({\mathcal {T}}u ={\mathcal {T}}v,\) we have
Multiply this by \(r_{\lambda , \ell '} (G (u) - G (v))\) and integrate on \({\mathbb {R}}^4\) to get
where \(\langle a, b \rangle _{\lambda , \ell '} :=\int a (x) b (x) (1 + \lambda | x |^2)^{- \ell '} d x.\)
Consequently, since G is non-decreasing and \(\langle u - v, G (u) - G (v) \rangle _{\lambda , \ell '} \geqslant 0\), from (43) we get
But by substituting \(G (u) - G (v)\) in place of \({\mathfrak {v}}\) in (41) we obtain
So, using (44) in above yield
But due to the integration by parts we have
This gives
where \(r_{\lambda , \ell '}^2 (x) = (1 + \lambda | x |^2)^{- \ell '}\) and \(\nabla r_{\ell '}^2 (x) = - 2 \lambda \ell ' (1 + \lambda | x |^2)^{- (\ell ' + 1)} x\) and
Hence, substituting (47) into (46) give
Consequently, using (48) into (45) provides
By taking sufficiently small \(\lambda \), using (42) together with (49) we get \(\Vert u - v \Vert _{L^2_{\lambda , \ell '}}^2 \leqslant 0\). This implies \(u = v\) in \(L^2_{\lambda , \ell '} ({\mathbb {R}}^4)\) and hence the map \({\mathcal {T}}\) is 1-1.
To prove surjectivity let \(v \in {\mathcal {B}}\) and \(\{ v_n \}_n \subset C_c^2 ({\mathbb {R}}^4)\) such that
Let \(h_n :=(- \Delta + m^2) v_n\). Then it follows, from the first part of Proposition 1, that the elliptic PDE
admits a unique solution in \(C_{\ell }^2 ({\mathbb {R}}^4)\). Then we get
Next, we prove that \(\{ u_n \}_n\) forms a Cauchy sequence in \(L_{\lambda , \ell '}^2 ({\mathbb {R}}^4)\) for sufficiently small \(\lambda > 0\). By multiplying
by \(r_{\lambda , \ell '}^2 (G (u_n) - G (u_m))\) and integrate on \({\mathbb {R}}^4\) we get
Since G is increasing, \(\langle u_n - u_m, G (u_n) - G (u_m) \rangle _{\lambda , \ell '} \geqslant 0\). Thus, the above implies
Thus, taking \({\mathfrak {v}}= G (u_n) - G (u_m)\) in (41) (modified version for \(\lambda \)) yield
So the last two estimates together with the Cauchy-Schwartz inequality give
Consequently, the computation as in (49) gives
Substituting \({\mathcal {G}} *(G (u_n) - G (u_m))\) from (51) into (52) followed by the reverse triangle inequality yield
Since \(\Vert v_n - v \Vert _{L^2_{\ell '} ({\mathbb {R}}^4)} \rightarrow 0 {\text {as}} n \rightarrow \infty \) and G is bounded, for sufficiently small \(\lambda > 0\) we get that \(\{ u_n \}_n\) forms a Cauchy sequence in \(L_{\lambda , \ell '}^2 ({\mathbb {R}}^4)\). Since \(L_{\lambda , \ell '}^2 ({\mathbb {R}}^4)\) is complete, there exists \(L_{\lambda , \ell '}^2 ({\mathbb {R}}^4) \ni u = \lim _{n \rightarrow \infty } u_n\). Since G is bounded, \(G (u) = \lim _{n \rightarrow \infty } G (u_n)\) in \(L_{\lambda , \ell '}^2 ({\mathbb {R}}^4)\). Thus by taking limit \(n \rightarrow \infty \) in (50) we obtain the existence of \(u \in L_{\lambda , \ell '}^2 ({\mathbb {R}}^4)\) such that
So if we show that \(u \in {\mathcal {B}}\) then we are done but that is true because,
which is finite. Hence \(u \in {\mathcal {B}}\) and we finish the proof of bijectivity of \({\mathcal {T}}\). \(\square \)
Hence we know that \({\mathcal {T}}^{- 1}\) exists. Let \({{\mathcal {T}}^{- 1}} (V) = v\) for some \(v, V \in {\mathcal {B}}\). Then \(V ={\mathcal {T}} (v)\) and from (39), we have that
From here it is clear that, for \(V \in {\mathcal {B}}\),
Consequently, by the Minkowski inequality for integral we get
In our next result we show that \({\mathcal {T}}^{- 1}\) is continuous as well.
Lemma 3
The map \({\mathcal {T}}^{- 1}\) is continuous on \({\mathcal {B}}.\)
Proof
Let \(\{ w_n \}_n \subset {\mathcal {B}}\) be a sequence converging to some \(w \in {\mathcal {B}}\). Let us set \({\mathcal {T}}^{- 1} w_n =:{\bar{w}}_n\) and \({\mathcal {T}}^{- 1} w =:{\bar{w}}.\) We will show that \({\bar{w}}_n \rightarrow {\bar{w}} \) as \(n \rightarrow \infty \) in \({\mathcal {B}}.\) Note that, we have
The first claim in the current proof is that the sequence \(\{ {\bar{w}}_n \}_n\) is relatively compact in \({\mathcal {B}}\). In order to prove this, first we show that \(\{ {\bar{w}}_n \}_n\) is uniformly bounded. Since \(\{ w_n \}_n\) is convergent in \({\mathcal {B}}\) and \(\alpha \) is a constant, due to (54) it is sufficient to show the uniform boundedness property for \(\left\{ \int _{{\mathbb {R}}^4} {\mathcal {G}} (\cdot - y) K_R (\exp (\alpha {\bar{w}}_n (y))) \, \textrm{d}y \right\} _n \subset {\mathcal {B}}.\) For this observe that, by (47), (48) of [2] we have
where the rhs is bounded uniformly in x and n. To move further, let us set
But by its structure we know that \({\bar{g}}_n\) solves the following equation uniquely
Thus,
This further implies, due to embedding, see (3.10) in [41], \(B^2_{\infty , \infty , \ell } ({\mathbb {R}}^4) \hookrightarrow B^{1 / 2}_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) and the equivalency of \(B^{1 / 2}_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) with \(\frac{1}{2}\)-Hölder weighted continuous functions, the equicontinuity of \(\{ {\bar{g}}_n \}_n\). Thus, since the uniform topology, which space \({\mathcal {B}}\) has, implies the topology of compact convergence, the Ascoli–Arzelà theorem (e.g. see Theorem 47.1 on page 290 in [40]) implies the relative compactness of \(\{ {\bar{w}}_n \}_n \subset {\mathcal {B}}\). Let us denote a converging subsequence \(\{ {\bar{w}}_{n_k} \}_k\) of \(\{ {\bar{w}}_n \}_n\) and set the limit as \({\mathcal {B}} \ni {\hat{w}} :=\lim _{k \rightarrow \infty } {\bar{w}}_{n_k}\). Since \(G (\cdot ) = \alpha K_R (\exp (\alpha \cdot - C_{\varepsilon }))\) is smooth and bounded, we have
as \(k \rightarrow \infty \) and thus passing the limit \(k \rightarrow \infty \) in (54) yield
But since \({\mathcal {T}}^{- 1} w = {\bar{w}}\) and \({\mathcal {T}}\) is bijective, we have \({\hat{w}} = {\bar{w}}.\) Consequently, any converging subsequence \(\{ {\bar{w}}_{n_k} \}_k\) converges to \({\bar{w}}\), which implies the continuity as desired. Hence the proof of continuity of \({\mathcal {T}}^{- 1}\) on \({\mathcal {B}}\) is complete. \(\square \)
Recall that we aim to prove that, for fixed \(\varepsilon \) and R, \(\varphi _{\varepsilon , R}\), which solves (36) and has representation (38), is Malliavin differentiable. Due to (40), in order to prove the differentiability of \(\varphi _{\varepsilon , R}\) or say F as the next step we show that the map \({\mathcal {T}}^{- 1}\), whose existence and continuity is proved, respectively, in Lemmata 2 and 3, is differentiable.
Lemma 4
The map \({\mathcal {T}}^{- 1}\) is differentiable and there exists a constant \(M > 0\) (depends on m) such that
where \({\mathfrak {L}} ({\mathcal {B}}, {\mathcal {B}})\) is the set of all bounded linear operators from \({\mathcal {B}}\) to \({\mathcal {B}}\), uniformly for \(v \in {\mathcal {B}}\).
Proof
It is straightforward to see that the Gateaux derivative of \({\mathcal {T}}\) at \(v \in {\mathcal {B}}\) in the direction of an arbitrary \(w \in {\mathcal {B}}\), is
where recall that \(G (v (\cdot )) = \alpha K_R (\exp (\alpha v (\cdot ) - C_{\varepsilon })).\) Thus, \({\mathcal {T}}\) is differentiable. Let us denote by \({\mathcal {T}}_v' (w)\) the derivative of \({\mathcal {T}}\) at \(v \in {\mathcal {B}}\) in the direction of \(w \in {\mathcal {B}}\) which is defined above, i.e.,
Note that since \(G'\) is bounded and \(w \in {\mathcal {B}}\), \({\mathcal {T}}_v' (w)\) is a well-defined element of \({\mathcal {B}}\). Next, let us fix \(v \in {\mathcal {B}}\) in the remaining part of the proof.
We claim that \({\mathcal {T}}_v'\) is one-one. Indeed, let \({\mathcal {T}}_v' (w) = 0\) for each \(w \in {\mathcal {B}}\) as element of \({\mathcal {B}}\) then by (55) we deduce that w solves the following equation
It is clear that \(w = 0\) is a solution to (56). From the computation in the proof of Proposition 1 and Lemma 5, we know that (56) has a unique solution in \(C_{\ell }^2 ({\mathbb {R}}^4)\), in particular \(w = 0\) is the unique solution to (56). Thus, \({\mathcal {T}}_v'\) is non-degenerate and \(({\mathcal {T}}_v')^{- 1} \in {\mathfrak {L}} ({\mathcal {B}}, {\mathcal {B}})\) is well-defined.
We aim to show that \(({\mathcal {T}}_v')^{- 1} \in {\mathfrak {L}} ({\mathcal {B}}, {\mathcal {B}})\) is uniformly bounded in \(v \in {\mathcal {B}}\). For this let us take any \(U \in {\mathcal {B}}\) and \(W :=({\mathcal {T}}_v')^{- 1} (U)\). Note that \(({\mathcal {T}}_v')^{- 1} (U)\) satisfies
Let us consider \(C_{c, \ell }^2 ({\mathbb {R}}^4)\), space of functions in \(C_{\ell }^2 ({\mathbb {R}}^4)\) having compact support, as subset of \({\mathcal {B}}\). Let \(U \in C_{c, \ell }^2 ({\mathbb {R}}^4)\) and let \(V = (- \Delta + m^2) U \in {\mathcal {B}}\). Then from (57) we get that \(({\mathcal {T}}_v')^{- 1} (U)\) satisfy the following equation
Here \(G' (v) ({\mathcal {T}}_v')^{- 1} (U)\) is simply the product of two functions \(G' (v)\) and \(({\mathcal {T}}_v')^{- 1} (U)\). Thus, since \(G' \geqslant 0\), Theorem 5.1 on page 145 in [22] implies, for \(w \in {\mathcal {B}}\),
by the Feynman–Kac formula, where B is an \({\mathbb {R}}^4\)-valued Brownian motion which starts at x defined on a complete probability space \(({\tilde{\Omega }}, \tilde{{\mathfrak {F}}}, \tilde{{\mathbb {P}}})\) and \(\tilde{{\mathbb {E}}}_x\) denotes the expectation w.r.t. \(\tilde{{\mathbb {P}}}\). But the r.h.s. in above can be estimated as follows to get, since \(\exp \left( - \int _0^t G' (v (B_s)) \, \textrm{d}s \right) \) is bounded,
This gives
which is finite. Consequently, by extension to \({\mathcal {B}}\), we get that there exists a constant \(M > 0\) (depends on m) such that
\(\square \)
Now we come to an important result of our paper that justifies the Malliavin differentiability of \(\varphi _{\varepsilon , R}\), for fix \(\varepsilon \) and R, which solves (36). In other words, the next result gives the differentiability of F which is defined in (38).
Theorem 3
Let us fix \(\varepsilon > 0\), \(R > 1\) and \(x \in {\mathbb {R}}^4\). The solution \(\varphi :=\varphi _{\varepsilon , R}\) to (36) is such that \(\varphi (y) \in {\mathbb {D}}^{1, 2}\), for every \(y \in {\mathbb {R}}^4\). Moreover, the process \(\{ D_z \varphi (x), z \in {\mathbb {R}}^4 \}\) satisfies
which is equivalent to
Proof
Since \(\varepsilon > 0\) and \(R > 1\) are fixed, we will avoid there explicit dependency. The idea of the proof is to show that the conditions of Theorem 2 with \(F (\omega ) :=\varphi _{\varepsilon , R} (x, \omega )\), are satisfied which will imply the conclusions of the current result. By (53) we have that that \(\varphi _{\varepsilon , R} (x) \in L^2 (\Omega )\). Indeed,
which is finite with the bound depends on R. Denote by \({\mathfrak {G}}h\) the following defined function
Note that \(\varphi _{0, \varepsilon } (\omega ) ={\mathfrak {G}} \xi \). Observe that, for \(h \in L^2_{\ell } ({\mathbb {R}}^4)\), due to (40)
But, since the above expression is linear in t, \(F [\omega + t i (h)]\) as function of t is an absolutely continuous function of t. From Lemma 4, we know that \({\mathbb {P}}\)-a.s. \({\mathcal {T}}_{\varphi _{0, \varepsilon }}'\) exists and non-degenerate and satisfy \(\Vert ({\mathcal {T}}_{\varphi _{0, \varepsilon }}')^{- 1} \Vert _{{\mathfrak {L}} ({\mathcal {B}}, {\mathcal {B}})} \leqslant M\), \({\mathbb {P}}\)-a.s. Thus, \({\mathbb {P}}\)-a.s. we have
Finally, due to the nice decay property of \({\mathfrak {a}}_{\varepsilon } *{\mathcal {G}}\) and Hölder inequality, \({\mathbb {P}}\)-a.s. we have
but this is finite. Now we define an H-valued random variable by
This is well-defined by the Riesz representation theorem and satisfies point (2) of Theorem 2. Hence, we complete the proof of Theorem 3. \(\square \)
Recall that \(\varphi _{\varepsilon , R}\) is the unique solution to (36). Proceeding further, we set \(\theta _{\varepsilon , R}^z :=D_z (\varphi _{\varepsilon , R})\). By applying Theorem 3 to \(\varphi _{\varepsilon , R}\), we ascertain that \(\varphi _{\varepsilon , R} (x) \in {\mathbb {D}}^{1, 2}\) and at point \(z \in {\mathbb {R}}^4,\)
Here we have also used the chain rule (Proposition 1.2.3 of [44]). Since (58) is linear in \(\theta _{\varepsilon , R}^z\), the Feynman-Kac formula yields
where
Here \({\mathbb {E}}_x\) is the expectation operator w.r.t. the probability measure \({\mathbb {P}}^x\) and {\(B_t, t \geqslant 0\)} is a \({\mathbb {R}}^4\)-valued Brownian motion under \({\mathbb {P}}^x\) with initial condition \(B_0 = x\). Observe that, for \(x_1, x_2 \in {\mathbb {R}}^4,\) expressions (34) followed by (33) yield
Since, see Corollary B.6 in [21] for the proof, for every \(F \in L^2 (\Omega )\) we have
we estimate the second term in the r.h.s. of (60), by the Hölder inequality and Proposition 1.2.3 of [44] along with Lipschitzness of \(F_1\) and \(F_2\), as
Further, since \((I - L)^{- 1}\) is a bounded operator on \(L^2 (\Omega )\), as in (62), the first integral in r.h.s. of (60) satisfies
Thus, substituting (62)–(63) into (60) yield
Applying the Minkowski inequality for integrals and the Fubini theorem to (59) we further have
Note that the r.h.s. of (65) is independent of R. Hence, substituting the above into (64) and using the Feynman-Kac formula (59) (with zero nonlinearity), we obtain
To estimate I(x, y), we utilize the following representation of the kernel \({\mathcal {G}}\), which is based on the Fourier transform, see pg. 273 of [7],
Thus, we deduce that, for all \(x, y \in {\mathbb {R}}^4\),
Next, we apply a change of variable \(z \mapsto A z\), in (67), where A represents the rotation matrix on \({\mathbb {R}}^4\) such that the vector \(x - y \in {\mathbb {R}}^4_H\) transform to align with one axis, let’s say the first axis. Then, with \(z =A w\), we have
where
Let us only consider the positive sign in \(e^{\pm i w_1}\). A similar approach will handle the negative sign case, with the contour C containing \(-i\) instead i.
First, we compute the integral \(\int _{{\mathbb {R}}} \frac{e^{i w_1} (({\mathcal {F}} ({\mathfrak {a}}_{\varepsilon })) (A (w_1, w_{1, \perp })))^2}{(m^2 | x - y |^2 + w_{^1}^2 + | w_{1, \perp } |^2)^2} \textrm{d}w_1\) for fixed \(w_{1, \perp } \in {\mathbb {R}}^3\) using the residue theorem. We define the contour C that traverses along the real lime from \(- a\) to a and then counterclockwise along a semicircle centered at 0 from \(- a\) to a. Choosing \(a \ge 1\) ensures that the point \(i (m^2 | x - y |^2 + | w_{1, \perp } |^2)^{1 / 2}\) lies within the contour. Now, consider the contour integral
Since the integrand in (68) has singularities at \(\pm i (m^2 | x - y |^2 + | w_{1, \perp } |^2)^{1 / 2}\) with multiplicity 2, by the residue theorem, for fixed \(w_{1, \perp } \in {\mathbb {R}}^3\), we have
where
and
Here, with abbreviated notation \({\mathcal {F}} ({\mathfrak {a}}_{\varepsilon }) = ({\mathcal {F}} ({\mathfrak {a}}_{\varepsilon })) (A (w_1, w_{1, \perp }))\), we find that
Hence, substituting the above expression in (70) and then taking the limit as \(a \rightarrow \infty \) in (69), we obtain, for fixed \(w_{1, \perp } \in {\mathbb {R}}^3\),
where we set \(m^2 | x - y |^2 + | w_{1, \perp } |^2 =:{\mathfrak {w}}_{1, \perp } (x, y).\) Consequently,
Since \(- i x_1 {\mathfrak {a}}_{\varepsilon } (x)\), where \(x= (x_1,x_2,x_3,x_4)\), is also a smooth and compactly supported function, it suffices to estimate \(I_1\) and \(I_3\) in (71). For \(I_1\), using estimate (81), for \(N =1\), where we let \(w_{1, \perp } = m u | x - y |\), we have
For \(I_3\) in (71), we can perform similar computation and obtain,
Thus, substituting (72)–(73) into (71), yields
Further, by substituting this into (66), we can make the estimation under the condition \(l = | x_1 - x_2 | > 2\) as
The complementary case of \(| x_1 - x_2 | \leqslant 2\) is straightforward, akin to the coupling approach. Therefore, since (74) holds uniformly in \(\varepsilon \) and R, we conclude the proof of Theorem 1 by first taking the limit taking \(R \rightarrow \infty \) and then letting \(\varepsilon \rightarrow 0\).
References
Albeverio, S., De Vecchi, F.C., Gubinelli, M.: Elliptic stochastic quantization. Ann. Probab. 48(4), 1693–1741 (2020)
Albeverio, S., De Vecchi, F.C., Gubinelli, M.: The elliptic stochastic quantization of some two dimensional Euclidean QFTs. Ann. Inst. Henri Poincaré Probab. Stat. 57(4), 2372–2414 (2021)
Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time. J. Funct. Anal. 16, 39–82 (1974)
Albeverio, S., Høegh-Krohn, R., Zegarliński, B.: Uniqueness of Gibbs states for general \(P (\varphi )_2\)-weak coupling models by cluster expansion. Commun. Math. Phys. 121(4), 683–697 (1989)
Albeverio, S., Kusuoka, S.: The invariant measure and the flow associated to the \(\Phi ^4_3\)-quantum field model. Ann. Sci. Norm. Super. Pisa Cl. Sci. 20(4), 1359–1427 (2020)
Abdesselam, A., Procacci, A., Scoppola, B.: Clustering bounds on \(n\)-point correlations for unbounded spin systems. J. Stat. Phys. 136(3), 405–452 (2009)
Albeverio, S., Yoshida, M.W.: \(H-C^1\) maps and elliptic SPDEs with non-linear local perturbations of Nelson’s Euclidean free field. J. Funct. Anal. 196(2), 265–322 (2002)
Barashkov, N.: A stochastic control approach to Sine Gordon EQFT. Ar**v:2203.06626, mar (2022)
Bauerschmidt, R., Bodineau, T.: Log-Sobolev inequality for the continuum Sine-Gordon model. Commun. Pure Appl. Math. 74(10), 2064–2113 (2021)
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Fundamental Principles of Mathematical Sciences. Springer, Heidelberg (2011)
Bauerschmidt, R., Dagallier, B.: Log-Sobolev inequality for the \(\varphi ^4_2\) and \(\varphi ^4_3\) measures. Commun. Pure Appl. Math. 77, 2579–2612 (2024)
Barashkov, N., Gubinelli, M.: A variational method for \(\Phi ^4_3\). Duke Math. J. 169(17), 3339–3415 (2020)
Barashkov, N., Gubinelli, M.: On the variational method for Euclidean quantum fields in infinite volume. Probab. Math. Phys. 4(4), 761–801 (2023)
Barashkov, N., De Vecchi, F.C.: Elliptic stochastic quantization of Sinh-Gordon QFT. Ar**v:2108.12664 (2021)
Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation. Ann. Probab. 46(5), 2621–2679 (2018)
Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)
De Vecchi, C., Fresta, F.L., Gubinelli, M.: A stochastic analysis of subcritical Euclidean fermionic field theories. Ar**v:2210.15047 (2022)
Duminil-Copin, H., Goswami, S., Raoufi, A.: Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature. Commun. Math. Phys. 374(2), 891–921 (2018)
Dimock, J.: A cluster expansion for stochastic lattice fields. J. Statist. Phys. 58(5–6), 1181–1207 (1990)
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)
Furlan, M., Gubinelli, M.: Weak universality for a class of 3d stochastic reaction–diffusion models. Probab. Theory Related Fields 173(3–4), 1099–1164 (2019)
Friedman, A.: Stochastic Differential Equations and Applications, vol. 1. Academic Press, New York-London (1975)
Funaki, T.: The reversible measures of multi-dimensional Ginzburg-Landau type continuum model. Osaka J. Math. 28(3), 463–494 (1991)
Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, New York (1987)
Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupled p(\(\phi \))2 model and other applications of high temperature expansions. In: Wightman, A.S. (ed.) Part II: The Cluster Expansion, in Constructive Quantum Field Theory, Springer Lecture Notes in Physics Volume 25, Springer, Berlin (1973)
Glimm, J., Jaffe, A., Spencer, T.: The wightman axioms and particle structure in the p(\(\phi \))2 quantum field model. Ann. Math. 100(2), 585–632 (1974)
Gubinelli, M., Hofmanová, M.: Global solutions to elliptic and parabolic \(\phi ^4\) models in Euclidean space. Commun. Math. Phys. 368(3), 1201–1266 (2019)
Gubinelli, M., Hofmanová, M.: A PDE construction of the Euclidean \(\phi _3^4\) quantum field theory. Commun. Math. Phys. 384(1), 1–75 (2021)
Gubinelli, M., Meyer, S.-J.: The FBSDE approach to sine-Gordon up to \(6 \pi \). Ar**v:2401.13648 (2024)
Duch, P., Gubinelli, M., Rinaldi, P.: Parabolic stochastic quantisation of the fractional \(\Phi ^4_3\) model in the full subcritical regime. (2024)
Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)
Hepp, K.: On the connection between the LSZ and Wightman quantum field theory. Commun. Math. Phys. 1, 95–111 (1965)
Høegh-Krohn, R.: A general class of quantum fields without cut-offs in two space-time dimensions. Commun. Math. Phys. 21, 244–255 (1971)
Jaffe, A.: Constructive quantum field theory. In: Kibble, T. (ed.) Mathematical Physics. World Scientific, Singapore (2000)
Janson, S.: Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129. Cambridge University Press, Cambridge (1997)
Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409–436 (1985)
Jona-Lasinio, G., Mitter, P.K.: Large deviation estimates in the stochastic quantization of \(\phi ^{4}_{2}\). Commun. Math. Phys. 130(1), 111–121 (1990)
Jona-Lasinio, G., Sénéor, R.: Study of stochastic differential equations by constructive methods. I. J. Stat. Phys. 83(5–6), 1109–1148 (1996)
Kupiainen, A.: Renormalization group and stochastic pdes. Ann. Henri Poincaré 17(3), 497–535 (2016)
Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Hoboken (2000)
Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic \(\Phi ^4\) model in the plane. Ann. Probab. 45(4), 2398–2476 (2017)
Moinat, A., Weber, H.: Space-time localisation for the dynamic \(\Phi ^4_3\) model. Commun. Pure Appl. Math. 73(12), 2519–2555 (2020)
Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)
Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications, 2nd edn. Springer, Berlin (2006)
Parisi, G.: Correlation functions and computer simulations. Nucl. Phys. B 180(3), 378–384 (1981)
Parisi, G., Sourlas, N.: Random magnetic fields, supersymmetry, and negative dimensions. Phys. Rev. Lett. 43(11), 744–745 (1979)
Parisi, G., Sourlas, N.: Supersymmetric field theories and stochastic differential equations. Nucl. Phys. B 206(2), 321–332 (1982)
Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sinica 24(4), 483–496 (1981)
Tindel, S.: Quasilinear stochastic elliptic equations with reflection: the existence of a density. Bernoulli 4(4), 445–459 (1998)
Triebel, H.: Theory of function spaces. III. Monographs in Mathematics. BirkhäuserVerlag, Basel (2006)
Acknowledgements
Most of this work was written when the last author was a post-doctoral fellow at Universität Bielefeld, Germany. The financial support from the German Science Foundation DFG through the Research Unit FOR 2402 is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A Auxiliary results
Appendix A Auxiliary results
The first result in this section is about the solution theory to Eq. (35).
Proposition 1
For given \(\varepsilon \in (0, 1)\) and \(R \geqslant 1\), there exists a \({\bar{\varphi }}_{\varepsilon , R} \in C_{\ell }^2 ({\mathbb {R}}^4) :=B^2_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) which solves (35). Moreover, \(\alpha {\bar{\varphi }}_{\varepsilon , R} \leqslant 0.\)
Proof
Let us introduce the following map
We first show that there exists a solution \({\bar{\varphi }}_{\varepsilon , R} \in B^2_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) to the equation
We aim to use Schaefer’s fixed-point theorem (see Theorem 4 in Section 9.2 of Chapter 9 of [20]) to prove the claim. In order to do this we have to prove that \({\mathcal {K}}\) is continuous in \({\bar{\varphi }}_{\varepsilon , R}\), that it maps any bounded set into a compact set and that the set of solutions to the equations
is bounded uniformly for all \(0 \leqslant \lambda \leqslant 1\). The continuity of \({\mathcal {K}}\) is an easy consequence of continuity of \((- \Delta + m^2)^{- 1}\) from \(B^0_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) into \(B^2_{\infty , \infty , \ell } ({\mathbb {R}}^4)\) and properties of functions \(K_R\) and \(\exp \). The map \({\mathcal {K}}\) is compact because the Schauder estimates and embedding imply
and the immersion is compact, see Proposition 52 of [2]. Finally the uniform boundedness in \(\lambda \) follows from inequality (76). Thus, by Schaefer’s fixed-point theorem there exists a fixed point of \({\bar{\varphi }} ={\mathcal {K}} ({\bar{\varphi }}, \eta )\) in \(B^{2 - \delta }_{\infty , \infty , \ell + \delta '} ({\mathbb {R}}^4)\). Let us call it \({\bar{\varphi }}_{\varepsilon , R}\). Further note that, since \({\bar{\varphi }}_{\varepsilon , R}\) is a fixed point, (76) also give
Thus, \({\bar{\varphi }}_{\varepsilon , R} \in B^2_{\infty , \infty , \ell } ({\mathbb {R}}^4)\). Hence the first part of the proof.
Next, since \({\bar{\varphi }}_{\varepsilon , R} \in C_{\ell }^2 ({\mathbb {R}}^4)\), \({\bar{\varphi }}_{\varepsilon , R} \in L_{\ell }^{\infty } ({\mathbb {R}}^4)\). Let us define, for \(x \in {\mathbb {R}}^4\),
where \(\ell \) is chosen such that the first part of the current proposition holds valid. Note that from the first part, \(\psi \) is bounded and locally belongs to \(C^2 ({\mathbb {R}}^4)\). Assume for the moment that \(\psi \) has a global maximum and attains its maximum value at \({\hat{x}}.\) Then, since \({\hat{x}}\) is a critical point,
and, thus, by the second derivative test,
But
so, from (77)
Since \(\alpha ^2 K_R (\exp (\alpha {\bar{\varphi }}_{\varepsilon , R}) \eta _{\varepsilon }) \geqslant 0\), we get
But note that due to the choice of weight \(r_{\ell , \theta }\), we can choose \(\theta > 0\) such that
Hence, with the choice of \(\theta \) from (78) we have
where the quantity in curly bracket is positive. Thus, the above is only possible if \(\alpha {\bar{\varphi }}_{\varepsilon , R} \leqslant 0\). Hence we have prove the result in the case \({\bar{\varphi }}_{\varepsilon , R}\) attains its maximum. The case when it does not, can be taken care as explained in Lemma 2.8 in [27]. This completes the proof. \(\square \)
Now we move to the uniqueness of above constructed solution. The approach to prove the next result is closely related to the proof of Lemma 31 in [2].
Lemma 5
For given \(\varepsilon \in (0, 1)\) and \(R \geqslant 1\), the solution to Eq. (35) is unique in \(C_{\ell }^2 ({\mathbb {R}}^4)\).
Proof
Let \(\varepsilon \in (0, 1)\) and \(R \geqslant 1\) be fixed parameters. We will omit explicit mention of them for the remainder of the proof. Consider \(J: {\mathbb {R}} \rightarrow {\mathbb {R}}\), a smooth, bounded, strictly increasing function such that \(J (0) = 0\) and \(J (- x) = - J (x)\). Further, let \({\bar{\varphi }}_1\) and \({\bar{\varphi }}_2\) be two solutions to equation (35). Since they are smooth, \(J ({\bar{\varphi }}_1 - {\bar{\varphi }}_2) \in C_{\ell }^2 ({\mathbb {R}}^4)\) implying that \(r_{\ell '} (\lambda z) J ({\bar{\varphi }}_1 - {\bar{\varphi }}_2) \in C_{\ell }^2 ({\mathbb {R}}^4)\) for \(\ell ' > 0\) sufficiently large enough and any \(\lambda > 0\). This implies that, where \(\langle \cdot , \cdot \rangle \) denotes just the \(L^2 ({\mathbb {R}}^4)\)-inner product,
where \({\mathcal {K}}\) is defined in (75).
We claim that the inequality
holds for sufficiently small \(\lambda > 0\) and some constant \(C > 0\). Indeed, we have
where \(J^{- 1} (t) = \int _0^t J (\tau ) \textrm{d}\tau \). By selecting a sufficiently small \(\lambda > 0\), we get the claim. For the first inequality we utilize the following fact:
which holds true since \(J^{- 1}\) is a Lipschitz function satisfying \(J^{- 1} (0)= 0\). Additionally, we exploit the increasing behavior of J to establish \(J^{- 1} (t) \leqslant t J (t) \).
The next claim is that \(\langle r_{\ell '} (\lambda z) J ({\bar{\varphi }}_1 - {\bar{\varphi }}_2), (- \Delta + m^2) (-{\mathcal {K}}({\bar{\varphi }}_1, \eta ) +{\mathcal {K}}({\bar{\varphi }}_2, \eta )) \rangle \geqslant 0\). To demonstrate this, we have
where we use the fact that \((\alpha (K_R (\exp (\alpha t_1) \eta )) - \alpha (K_R (\exp (\alpha t_2) \eta ))) \cdot J (t_1 - t_2)\) is positive since both \(\alpha (K_R (\exp (\alpha \cdot ) \eta ))\) and J are increasing functions and \(J (0) = 0\).
Thus, combining inequalities (79) and (80) we deduce that
which implies \({\bar{\varphi }}_1 - {\bar{\varphi }}_2 = 0\), since J is a strictly increasing function. Consequently, the proof of uniqueness is established. \(\square \)
For the next result assume that \(\Delta \) is the d-dimensional Laplacian. Let us denote the kernel representation of \({\mathcal {L}}^{- 1}\) by
Lemma 6
\({\mathcal {G}}\) has the following integral representation
Moreover, there exist some constants \(C_1, C_2 > 0\) such that the following holds:
-
1.
if \(d > 2\) then
$$\begin{aligned} {\mathcal {G}} (x) \le C_1 | x |^{- d + 2} { {\text {if}}} | x | < 1 {\text { and }} \qquad C_1 e^{- C_2 | x | } \quad \text { if} | x | \ge 1; \end{aligned}$$ -
2.
if \(d < 2\) then
$$\begin{aligned} {\mathcal {G}} (x) \le C_1 | x |^{- d + 2} \text { {for} } x \in {\mathbb {R}}^d; \end{aligned}$$ -
3.
if \(d = 2\) then
$$\begin{aligned} {\mathcal {G}} (x) \le C_1 - \frac{2}{(4 \pi )^{\frac{d}{2}} \Gamma \left( \frac{d}{2} \right) } \log (| x |) \text { if } | x | < 1 { \text { and }} \qquad C_1 e^{- C_2 | x | } \quad {{\text { if }}} | x | \ge 1. \end{aligned}$$
Proof
See Proposition A.1 in [7]. \(\square \)
The next result is well-known in the literature.
Theorem 4
(Paley–Wiener–Schwartz) For any \(d \in {\mathbb {N}}\), the vector space \(C_c^{\infty }({\mathbb {R}}^d)\), comprising compactly supported smooth functions on \({\mathbb {R}}^d\), is isomorphic, via the Fourier transform, to the space of entire functions F on \({\mathbb {C}}^d\) satisfying the following condition: there exists a positive real number B such that for every integer \(N > 0\), there is a real number \(C_N > 0\) such that
This implies that for any \(u \in C_c^{\infty } ({\mathbb {R}}^d)\), there exists an entire function \(F = {\hat{u}}\) satisfying the above estimate.
Finally we need the following Besov embedding.
Theorem 5
Consider \(p_1, p_2, q_1, q_2 \in [1, \infty ], s_1 > s_2\) and \(\ell _1, \ell _2 \in {\mathbb {R}}\) such that
then \(B_{p_1, q_1, \ell _1}^{s_1} ({\mathbb {R}}^d)\) is continuously embedded in \(B_{p_2, q_2, \ell _2}^{s_2} ({\mathbb {R}}^d)\). And if \(\ell _1 < \ell _2\) and \(s_1 - \frac{d}{p_1} > s_2 - \frac{d}{p_2}\) then the embedding \(B_{p_1, q_1, \ell _1}^{s_1} ({\mathbb {R}}^d) \hookrightarrow B_{p_2, q_2, \ell _2}^{s_2} ({\mathbb {R}}^d)\) is compact.
Proof
See Theorem 6.7 in [50]. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gubinelli, M., Hofmanová, M. & Rana, N. Decay of correlations in stochastic quantization: the exponential Euclidean field in two dimensions. Stoch PDE: Anal Comp (2024). https://doi.org/10.1007/s40072-024-00328-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40072-024-00328-x