  In this paper we study the existence and uniqueness of the
$L_{\rho}^{2p}(\mathbb{R}^d;\mathbb{R}^1)\times
L_{\rho}^2(\mathbb{R}^d;\mathbb{R}^d)$ valued solution of backward doubly
stochastic differential equations with polynomial growth coefficients using
week 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 backward doubly stochastic differential equations (BDSDEs).
This probabilistic representation is then used to prove the existence of
stationary solution of SPDEs on $\mathbb{R}^d$ via infinite horizon BDSDEs. The
convergence of the solution of BDSDE to the solution of infinite horizon BDSDEs
is shown to be equivalent to the convergence of the pull-back of the solutions
of SPDEs. With this we obtain the stability of the stationary solutions as
well.
