  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.
