The parametrix method for parabolic SPDEs

Abstract

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Itô–Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

Introduction

Stochastic partial differential equations (SPDEs) arise in many applications in probability theory and in particular in the study of filtering problems (see e.g. [8], [15]). Assume that $(X_t,Y_t)$ is a diffusion where $X$ represents a signal that is not directly observable and we want to extract information about $X$ from ${\mathcal{F}}_t^Y=\sigma (Y_s,s\le t)$ that is the filtration of the observations on $Y$. Then, under natural assumptions, for any bounded and measurable function $f$ we have

$$E\left[f\left(X_t\right)\mid {\mathcal{F}}_t^Y\right]={\int }_{{\mathbb{R}}^d}f\left(x\right)p_t\left(x\right)dx$$

where $p_t\left(x\right)$ denotes the conditional density of $X_t$ given ${\mathcal{F}}_t^Y$: it turns out that $p_t$ satisfies a SPDE of the form

$$d{p}_t\left(x\right)={\mathbf{L}}_t{p}_t\left(x\right)dt+{\mathbf{G}}_t{p}_t\left(x\right)d{W}_t,$$

where ${\mathbf{L}}_t$ is a second-order elliptic operator and ${\mathbf{G}}_t$ is a first-order operator. The coefficients of ${\mathbf{L}}_t$ and ${\mathbf{G}}_t$ may depend on $t,x,Y_t$ and are therefore random and typically not smooth. A very particular case is when $Y\equiv 0$: then ${\mathbf{G}}_t=0$ and (1.1) reduces to the classical forward Kolmogorov equation for the transition density $p_t$ of $X_t$. In the general case, $p_t$ can be referred to as the stochastic fundamental solution of (1.1). The aim of this paper is to prove existence, regularity and estimates from above and below of $p_t$ by using a classical tool from PDEs’ theory, the parametrix method.

Let $(\Omega ,\mathcal{F},P)$ be a complete probability space with an increasing filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$ of complete with respect to $(\mathcal{F},P)$ $\sigma$-fields ${\mathcal{F}}_t\subseteq \mathcal{F}$. Let $d_1\in \mathbb{N}$ and let $W^k$, $k=1,\dots ,d_1$, be one-dimensional independent Wiener processes with respect to ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$. We consider the parabolic SPDE

$$d{u}_t\left(x\right)=\left({\mathbf{L}}_t{u}_t\left(x\right)+{\mathbf{f}}_t\left(x\right)\right)dt+{\mathbf{G}}_{{\sigma }_t^k}{u}_t\left(x\right)d{W}_t^k,$$

where ${\mathbf{L}}_t$ is the second-order operator

$${\mathbf{L}}_t{u}_t\left(x\right)=\frac{1}{2}{\mathbf{a}}_t^{ij}\left(x\right){\partial }_{ij}{u}_t\left(x\right)+{\mathbf{b}}_t^j\left(x\right){\partial }_j{u}_t\left(x\right)+{\mathbf{c}}_t\left(x\right)u_t\left(x\right)$$

and ${\mathbf{G}}_{{\sigma }_t^k}$ is the first-order operator

$${\mathbf{G}}_{{\sigma }_t^k}{u}_t\left(x\right)={\sigma }_t^{ik}\left(x\right){\partial }_i{u}_t\left(x\right).$$

Throughout the paper, the summation convention over repeated indices is enforced regardless of whether they stand at the same level or at different ones. The point of ${\mathbb{R}}^d$ is denoted by $x=(x_1,\dots ,x_d)$ and we set ${\partial }_i={\partial }_{x_i}$, $\nabla =\left({\partial }_1,\dots ,{\partial }_d\right)$ and ${\partial }_{ij}={\partial }_i{\partial }_j$. A function $u=u_t(x,\omega )$ on $[0,\infty )\times {\mathbb{R}}^d\times \Omega$ is denoted by $u_t\left(x\right)$ and we shall systematically omit the explicit dependence on $\omega \in \Omega$. The coefficients ${\mathbf{a}}_t$, ${\mathbf{b}}_t$, ${\mathbf{c}}_t$, ${\mathbf{f}}_t$ and ${\sigma }_t^k$ in (1.2) are intended to be random and not smooth.

The Cauchy problem for evolution SPDEs has been studied by several authors. Under coercivity conditions analogous to uniform ellipticity for PDEs, there exists a complete theory in Sobolev spaces (see [6], [16], [20], [25] and references therein) and in the spaces of Bessel potentials (see [11] and [12]). Classical solutions in Hölder classes were first considered in [24], [26] and more recent results were proved in [1] and [18], though the authors only considered equations with non-random coefficients and with no derivatives of the unknown function in the stochastic term.

In the last decades, the use of analytical or PDE techniques in the study of SPDEs has become widespread. For instance, the results in [1], [18], [30] are based on classical methods of deterministic PDEs, such as Duhamel principle and a priori Schauder estimates; the $L^p$ estimates in [3] are proved by adapting the classical Moser’s iterative argument; [28] provides short-time asymptotics of random heat kernels. A further remarkable example is given by the recent series of papers by Krylov [13], [14], [15] where the Hörmander’s theorem for SPDEs is proved; see also the very recent results in [23] for backward SPDEs.

In this paper we extend another classical tool that, to the best of our knowledge, has not yet been considered in the study of SPDEs, the well-known parametrix method for the construction of the fundamental solution of PDEs with Hölder continuous coefficients. There are two main problems that one faces when trying to apply the parametrix method to SPDEs: the lack of a Duhamel principle for SPDEs and the roughness of the coefficients, that are assumed to be only measurable in time. In the first part of the paper, we use the Itô–Wentzell formula [29] to make a random change of variables which transforms the SPDE in a PDE with random coefficients; the latter admits a Duhamel principle and, in the second part of the paper, we use it to extend the parametrix method to parabolic PDEs with coefficients measurable in the time variable.

This paper does not pretend to encompass the most general assumptions but rather investigate the possible use of the parametrix method in the stochastic framework; as such, it has to be intended as a first step of a research programme aiming at considering more general classes of possibly degenerate SPDEs. More precisely, it is very likely that the techniques used in this paper can be applied to SPDEs satisfying the strong Hörmander condition, such as those considered in [13]. A more challenging problem is to consider the Langevin SPDE

$$d{u}_t(x,y)=\left(\frac{a_t(x,y)}{2}{\partial }_{xx}{u}_t(x,y)+x{\partial }_y{u}_t(x,y)\right)dt+{\sigma }_t(x,y){\partial }_x{u}_t(x,y)d{W}_t,(x,y)\in {\mathbb{R}}^2.$$

This equation has unbounded drift coefficient and satisfies the weak Hörmander condition. The parametrix method has been generalized to deterministic (i.e. with ${\sigma }_t\equiv 0$) Langevin PDEs in [4], [22] and [10]; however, contrary to the uniformly parabolic case considered in the present paper, the intrinsic Hölder regularity in the spatial variables cannot be studied independently from the time variable as it was recently shown in [19]. This is an additional issue that needs careful investigation and is the subject of the forthcoming paper [21].

The paper is organized as follows. In Section 2 we introduce the basic notations and state our main result, Theorem 2.5; for illustrative purposes, the particular case of the stochastic heat equation is discussed in Section 2.1. In Section 3 we recall the Itô–Wentzell formula and provide some estimate for the related flow of diffeomorphisms. In Section 4 we present the parametrix method. Since the complete proofs are rather technical and to a large extent similar to the classical case, we only provide the details on those aspects that require significant modifications: in particular, in Section 4.3 we present a proof of the Gaussian lower bound for the fundamental solution which requires some non trivial adaptation of an original argument by Aronson (cf. [5]).