Stationary Stochastic Viscosity Solutions of SPDEs

TL;DR

This paper develops a framework for finding stationary stochastic viscosity solutions of parabolic SPDEs using infinite horizon backward doubly stochastic differential equations, establishing existence, uniqueness, and regularity results.

Contribution

It introduces a novel approach linking infinite horizon BDSDEs to stationary solutions of SPDEs, including a 'perfection procedure' for solution refinement.

Findings

Established existence and uniqueness of solutions for the infinite horizon BDSDEs.

Constructed stationary stochastic viscosity solutions for the SPDEs.

Demonstrated the regularity properties of the solutions.

Abstract

In this paper we aim to find the stationary stochastic viscosity solutions of a parabolic type SPDEs through the infinite horizon backward doubly stochastic differential equations (BDSDEs). For this, we study the existence, uniqueness and regularity of solutions of the corresponding infinite horizon BDSDEs as well as the "perfection procedure" applied to the solutions of BDSDEs. At last the "perfect" stationary stochastic viscosity solutions of SPDEs constructed by solutions of corresponding BDSDEs are obtained.