Recursive estimation of the invariant measure of a diffusion process.

Summary The purpose of this thesis is to study an algorithm, simple to implement and recursive, allowing to compute the integral of a function with respect to the invariant probability of a process solution of a finite dimensional stochastic differential equation. The main assumption on these solutions (diffusions) is the existence of a Lyapounov function guaranteeing a stability condition. By the ergodic theorem we know that empirical measures of diffusion converge to an invariant measure. We study a similar convergence when the scattering is discretized by an Euler scheme of decreasing pitch. We prove that the weighted empirical measures of this scheme converge to the invariant measure of the scattering, and that it is possible to integrate exponential functions when the scattering coefficient is sufficiently small. Moreover, for a more restricted class of diffusions, we prove the almost certain convergence in Lp of the Euler scheme to the diffusion. We obtain convergence speeds for the weighted empirical measures and give the parameters allowing an optimal speed. We finish the study of this scheme when there are multiple invariant measures. This study is done in dimension 1, and allows us to highlight a link between Feller classification and Lyapunov functions. In the last part, we present a new adaptive algorithm allowing to consider more general problems such as Hamiltonian systems or monotone systems. It consists in considering the empirical measures of an Euler scheme built from a sequence of adapted random steps dominated by a sequence decreasing to 0.