  We study a class of backward doubly stochastic differential equations
(BDSDEs) involving martingales with spatial parameters, and show that they
provide probabilistic interpretations (Feynman-Kac formulae) for certain
semilinear stochastic partial differential equations (SPDEs) with space-time
noise. As an application of the Feynman-Kac formulae, random periodic solutions
and stationary solutions to certain SPDEs are obtained.
