2006/02/28 by Qi Zhang, Huaizhong Zhao · 74 citations
Economics, Econometrics and Finance · Engineering · Mathematics · #Connection (principal bundle) #Geometry #Initial value problem #Mathematical analysis #Mathematics #Nonlinear Differential Equations Analysis #Partial differential equation #Stability and Controllability of Differential Equations #Stochastic differential equation #Stochastic partial differential equation #Stochastic processes and financial applications #Uniqueness #math.DS #math.PR
paper · pdf · doi:10.1016/j.jfa.2007.06.019
published in Journal of Functional Analysis 252(1), 171-219 (Elsevier BV)
openalex publication_date 2007/09/05 · arxiv created 2008/11/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between Lρ2(ℝd;ℝ1) ⊗ Lρ2(ℝd;ℝd) 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.