Progressive probabilistic representation of nonlinear nonconservative PDEs and particle algorithms.

Summary In this thesis, we propose a progressive (forward) approach for the probabilistic representation of nonlinear and nonconservative Partial Differential Equations (PDEs), allowing to develop a particle-based algorithm to numerically estimate their solutions. The Nonlinear Stochastic Differential Equations of McKean type (NLSDE) studied in the literature constitute a microscopic formulation of a phenomenon modeled macroscopically by a conservative PDE. A solution of such a NLSDE is the data of a couple $(Y,u)$ where $Y$ is a solution of a stochastic differential equation (SDE) whose coefficients depend on $u$ and $t$ such that $u(t,cdot)$ is the density of $Y_t$. The main contribution of this thesis is to consider nonconservative PDEs, i.e. conservative PDEs perturbed by a nonlinear term of the form $Lambda(u,nabla u)u$. This implies that a pair $(Y,u)$ will be a solution of the associated probabilistic representation if $Y$ is still a stochastic process and the relation between $Y$ and the function $u$ will then be more complex. Given the law of $Y$, the existence and uniqueness of $u$ are proved by a fixed point argument via an original Feynmann-Kac formulation.