Stationary Solutions of SPDEs and Infinite Horizon BDSDEs

Abstract

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between $L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^1)\otimes L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^d)$ valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.

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• Pathwise Random Periodic Solutions of Stochastic Differential Equations
• Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding
• Backward doubly stochastic differential equations with weak assumptions on the coefficients
• Forward-backward doubly stochastic systems and classical solutions of path-dependent stochastic PDEs
• Forward-Backward Doubly Stochastic Differential Equations with Random Jumps and Stochastic Partial Differential-Integral Equations
• Backward doubly stochastic differential equations and SPDEs with quadratic growth
• The Equivalence between Uniqueness and Continuous Dependence of Solution for BDSDEs
• Comparison Theorem of Multi-dimensional Backward Doubly Stochastic Differential Equations on Infinite Horizon