  In this paper we construct the stationary weak solutions of parabolic SPDEs
by a general infinite horizon backward doubly stochastic differential equations
(BDSDEs for short) with non-degenerate terminal functions. For this, we first
study the existence, uniqueness and stationarity of solutions of such kind of
BDSDEs in weighted $L^p(dx)\bigotimes L^2(dx)$ space ($p>2$). Then the
corresponding stationary solutions of parabolic SPDEs can be obtained by the
connection between the solutions of BDSDEs in weighted $L^p(dx)\bigotimes
L^2(dx)$ space and the weak solutions of parabolic SPDEs. This result shows
that the stationary solutions of SPDEs can be constructed by their
corresponding infinite horizon BDSDEs with any Lipschitz continuous terminal
function.
