Weak backward error analysis for stochastic Hamiltonian Systems

This paper presents a study of the approximation error corresponding to a symplectic scheme of weak order one for a stochastic autonomous Hamiltonian system. A backward error analysis is done at the level of the Kolmogorov equation associated with the initial stochastic Hamiltonian system. An expansion of the weak error and expansions of the ergodic averages and of the invariant measures associated with the numerical scheme are obtained in terms of powers of the discretization step size and the solutions of the modified Kolmogorov equation.