A General Construction to Stationary Weak Solutions of Parabolic SPDEs

TL;DR

This paper introduces a general method for constructing stationary weak solutions of parabolic SPDEs using infinite horizon backward doubly stochastic differential equations, establishing existence, uniqueness, and stationarity in weighted spaces.

Contribution

The paper develops a novel approach linking infinite horizon BDSDEs to stationary weak solutions of parabolic SPDEs, broadening the construction methods for such solutions.

Findings

Established existence and uniqueness of solutions in weighted $L^p$ spaces.

Connected BDSDE solutions to weak solutions of SPDEs.

Constructed stationary solutions for SPDEs with Lipschitz terminal functions.

Abstract

In this paper we construct the stationary weak solutions of parabolic SPDEs by a general infinite horizon backward doubly stochastic differential equations (BDSDEs for short) with non-degenerate terminal functions. For this, we first study the existence, uniqueness and stationarity of solutions of such kind of BDSDEs in weighted $L^p(dx)\bigotimes L^2(dx)$ space ($p>2$). Then the corresponding stationary solutions of parabolic SPDEs can be obtained by the connection between the solutions of BDSDEs in weighted $L^p(dx)\bigotimes L^2(dx)$ space and the weak solutions of parabolic SPDEs. This result shows that the stationary solutions of SPDEs can be constructed by their corresponding infinite horizon BDSDEs with any Lipschitz continuous terminal function.