Stochastic partial differential equations with obstacle.

Summary This thesis deals with Quasilinear Stochastic Partial Differential Equations. It is divided into two parts. The first part deals with the obstacle problem for quasilinear stochastic partial differential equations and the second part is devoted to the study of quasilinear stochastic partial differential equations driven by a Brownian G-movement. In the first part, we first show the existence and uniqueness of an obstacle problem for quasilinear stochastic partial differential equations (in short OSPDE). Our method is based on analytical techniques coming from the parabolic potential theory. The solution is expressed as a pair (u,v) where u is a continuous predictable process that takes its values in a Sobolev space and v is a random regular measure satisfying the Skohorod condition. Then, we establish a maximum principle for the local solution of quasilinear stochastic partial differential equations with obstacle. The proof is based on a version of Itô's formula and estimates for the positive part of a local solution which is negative on the edge of the considered domain. The objective of the second part is to study the existence and uniqueness of the solution of stochastic partial differential equations directed by G-Brownian motion in the framework of a space with sublinear expectation. We establish an Itô formula for the solution and a comparison theorem.