On the probabilistic interpretation of some nonlinear partial differential equations.

Summary The approach adopted in this thesis is the following. Considering a nonlinear evolution equation, we try to associate a probability p on a space of trajectories such that : - either the marginals in time of p have densities with respect to the lebesgue measure in space which are weak solution of the evolution equation - or, in the case of dimension one of space, the distribution functions of the marginals are weak solution of the equation Once the probability p is obtained as the unique solution of a nonlinear martingale problem, we construct a system of n probabilistically interacting particles whose empirical measure converges to p when n tends to infinity. Such a convergence result, called chaos propagation, allows us to approach the solutions of the evolution equation by simulating the system of particles. For the most part, we treat parabolic type equations such as the viscous burgers equation and the porous media equation in this program. Under p, the canonical process on the space of continuous trajectories is then a nonlinear diffusion. We are also interested in a kinetic equation related to scalar conservation laws and work for that on a space of trajectories with jumps. Finally, we show on examples that it is not necessary to restrict ourselves to the natural case where the initial condition of the evolution equation is a probability or a probabilistic distribution function. It is possible to adapt the approach to take into account bounded signed measures or their distribution functions.