A relative compactness criterion in Wiener–Sobolev spaces and application to semi-linear Stochastic PDEs

We prove a relative compactness criterion in Wiener–Sobolev space which represents a natural extension of the compact embedding of Sobolev space H1 into L2, at the level of random fields. Then we give a specific statement of this criterion for random fields solutions of semi-linear stochastic partial differential equations with coefficients bounded in an appropriate way. Finally, we employ this result to construct solutions for semi-linear stochastic partial differential equations with distribution as final condition. We also give a probabilistic interpretation of this solution in terms of backward doubly stochastic differential equations formulated in a weak sense.