Stochastic Hamiltonian Systems: Exponential Convergence to the Invariant Measure, and Discretization by the Implicit Euler Scheme

In this paper we carefully study the large time behaviour of u(t, x, y) := Ex,y f(Xt, Yt)− ∫ f dμ, where (Xt, Yt) is the solution of a stochastic Hamiltonian dissipative system with non gbally Lipschitz coefficients, μ its unique invariant law, and f a smooth function with polynomial growth at infinity. Our aim is to prove the exponential decay to 0 of u(t, x, y) and all its derivatives when t goes to infinity, for all (x, y) in R. We apply our precise estimates on u(t, x, y) to analyze the convergence rate of a probabilistic numerical method based upon the implicit Euler discretization scheme which approximates ∫ f dμ.