SPDEs with Polynomial Growth Coefficients and Malliavin Calculus Method

TL;DR

This paper develops a framework for analyzing SPDEs with polynomial growth coefficients using backward doubly stochastic differential equations and Malliavin calculus, establishing existence, uniqueness, and stability of solutions.

Contribution

It introduces a new probabilistic representation of weak solutions of SPDEs with polynomial growth, linking them to BDSDEs and proving stationary solution existence.

Findings

Existence and uniqueness of solutions for SPDEs with polynomial growth coefficients.

A new probabilistic representation of weak solutions via BDSDEs.

Proof of stability and existence of stationary solutions on .

Abstract

In this paper we study the existence and uniqueness of the $L_{\rho}^{2p}(\mathbb{R}^d;\mathbb{R}^1)\times L_{\rho}^2(\mathbb{R}^d;\mathbb{R}^d)$ valued solution of backward doubly stochastic differential equations with polynomial growth coefficients using week convergence, equivalence of norm principle and Wiener-Sobolev compactness arguments. Then we establish a new probabilistic representation of the weak solutions of SPDEs with polynomial growth coefficients through the solutions of the corresponding backward doubly stochastic differential equations (BDSDEs). This probabilistic representation is then used to prove the existence of stationary solution of SPDEs on $\mathbb{R}^d$ via infinite horizon BDSDEs. The convergence of the solution of BDSDE to the solution of infinite horizon BDSDEs is shown to be equivalent to the convergence of the pull-back of the solutions of SPDEs. With this we obtain the stability of the stationary solutions as well.