BDSDEs with locally monotone coefficients and Sobolev solutions for SPDEs

In this paper, we obtain the existence and uniqueness result of the solutions to backward doubly stochastic differential equations (BDSDEs for short) with locally monotone coefficients. As an intermediate step, we also obtain an existence and uniqueness result for the solutions to BDSDEs with one kind of globally monotone coefficients. And then we give the probabilistic interpretation for the solutions in Sobolev spaces of quasilinear stochastic partial differential equations (SPDEs for short) in terms of these two classes of BDSDEs.