Approximations of the equilibrium distributions of some stochastic systems with McKean-Vlasov interactions.

Summary In this thesis we propose a numerical approximation for the equilibrium measure of a McKean Vlasov stochastic differential equation (SDE), when the drift coefficient is given by a function with ergodic properties, which is perturbed by a Lipschitzian nonlinear interaction function. We establish a theorem of existence and uniqueness of the equilibrium measure, as well the exponential convergence rate to this equilibrium. We apply the method based on the obtention of Wasserstein contractions using the random coupling variables, as suggested by Cattiaux-Gullin-Malrieu (2006) for the convex potential drift case. After, using the particle system, the chaos propagation property and Euler’s scheme to approximate the SDE, we estimate numerically the integral of every Lipschit function w. R. T. The measure at fixed time, with a time-uniform estimation error. Then, using this numerical estimation we approximate the integral w. R. T. The equilibrium measure. Finally, in the one-dimensional case, we provide numerical estimations for the density and the cumulative distribution function of the equilibrium measure. We use the algorithm proposed by Bossy-Talay (1996) and obtain the optimal rate convergence of the approximation in different norms.