Degenerate Parabolic Stochastic Partial Differential Equations

Abstract

In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.