Backward doubly stochastic differential equations Stochastic partial differential equations Pathwise stationary solution Random dynamical systems
The main purpose of this paper is to study the existence of stationary
solution for stochastic partial differential equations. We establish a new connection
between backward doubly stochastic differential equations on infinite time horizon
and the stationary solution of the SPDEs. For this we study the existence of
the solution of the associated BDSDEs on infinite time horizon and prove it is a
stationary viscosity solution of the corresponding SPDEs.