SPDEs with polynomial growth coefficients and the Malliavin calculus method

In this paper we study the existence and uniqueness of the Lρ2p(Rd;R1)×Lρ2(Rd;Rd) valued solutions of backward doubly stochastic differential equations (BDSDEs) with polynomial growth coefficients using weak convergence, equivalence of norm principle and Wiener–Sobolev compactness arguments. Then we establish a new probabilistic representation of the weak solutions of SPDEs with polynomial growth coefficients through the solutions of the corresponding BDSDEs. This probabilistic representation is then used to prove the existence of stationary solutions of SPDEs on Rd via infinite horizon BDSDEs. The convergence of the solution of a finite horizon BDSDE, when its terminal time tends to infinity, to the solution of the infinite horizon BDSDE is shown to be equivalent to the convergence of the pull-back of the solution of corresponding SPDE to its stationary solution. This way we obtain the stability of the stationary solution naturally.