Approximation of the invariant probability measure of stochastic Hamiltonian dissipative systems with non globally Lipschitz coefficients

In this paper we study the convergence rate of the implicit Euler scheme for the approximation of invariant measures of stochastic Hamiltonian dissipative systems with non globally Lipschitz coefficients such as described by Soize [15]. The technical difficulty of the analysis comes from the polynomial growth of the coefficients and the degeneracy of the infinitesimal generators of Hamiltonian dissipative diffusion processes. The results of this paper are quite new, in particular the convergence rate analysis of the implicit Euler scheme and the estimates on the exponential decay in time of the solution of some degenerate parabolic partial differential equations with non globally Lipschitz coefficients. The latter estimates ensure, e.g., that moments of Hamiltonian dissipative systems converge exponentially fast when time goes to infinity.