Weak solutions for SPDEs and Backward Doubly Stochastic Differential Equations

We give the probabilistic interpretation of the solutions in Sobolev spaces of parabolic semilinear stochastic PDEs in terms of Backward Doubly Stocha stic Differential Equations. This is a generalization of the Feynman Kac fo rmula. Wc also discuss linear stochastic PDEs in which the terminal value a nd the coefficients are distributions.