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Section: New Results

Diffusion limit for a stochastic kinetic problem

Participants : Arnaud Debussche, Erwan Faou.

In [29] we consider numerical approximations of stochastic differential equations by the Euler method. In the case where the SDE is elliptic or hypo-elliptic, we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every invariant measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the Euler scheme is exponentially mixing.