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Zhang, Qi; Zhao, Huaizhong

Stationary solutions of SPDEs and infinite horizon BDSDEs. (English) Zbl 1127.60059

J. Funct. Anal. 252, No. 1, 171-219 (2007).
Summary: In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between \(L^2_{\rho}(\mathbb R^d; \mathbb R^1) \otimes L^2_{\rho} (\mathbb R^d; \mathbb R^d)\) 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 obtaining 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.

Cited in 1 Review
Cited in 42 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

backward doubly stochastic differential equations; weak solutions; stochastic partial differential equations; stationary solution; random dynamical systems
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References:

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