Stationary Solutions of SPDEs and Infinite Horizon BDSDEs with Non-Lipschitz Coefficients

TL;DR

This paper establishes a connection between solutions of infinite horizon backward doubly stochastic differential equations (BDSDEs) and stationary solutions of stochastic partial differential equations (SPDEs), proving existence and uniqueness under monotonicity conditions without Lipschitz assumptions.

Contribution

It introduces a general theorem linking solutions of infinite horizon BDSDEs to stationary solutions of SPDEs, extending existence and uniqueness results beyond Lipschitz conditions.

Findings

Solutions of finite horizon BDSDEs correspond to initial value solutions of SPDEs.

Infinite horizon BDSDE solutions yield stationary solutions of SPDEs.

Existence and uniqueness are proved under monotonicity without Lipschitz conditions.

Abstract

We prove a general theorem that the $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.