A stochastic particle method with random weights for approximating statistical solutions of McKean-Vlasov-Fokker-Plank equations.

Summary Partial differential equations (PDE) with random initial condition are used in the modeling of some complex physical phenomena such as turbulence. The characterization of the solution law, or statistical solution, has been the subject of much theoretical work. However, it is often difficult to estimate the accuracy of the usual methods of simulation of the average solutions of the e. D. P, or moments of the statistical solution. This thesis consists of two parts: we start by presenting the theory of statistical solutions, in particular in the case of the vortex equation of an incompressible fluid in the plane. This example leads us to consider, in the second part of this thesis, the model problem of a Mckean-Vlasov equation with random initial condition. Assuming that the coefficients of the equation are lipschitzian and bounded, we show that it admits a unique statistical solution whose moments can be represented using a nonlinear diffusion process. We deduce from this interpretation a stochastic particle method for the simulation of the moments. Its originality is that the interaction weights between the particles are random variables, defined from non-parametric estimators of a regression function. Finally, we study the convergence speed (theoretical and numerical) of the method for different families of weights.