  We consider numerical approximations of stochastic differential equations by
the Euler method. In the case where the SDE is elliptic or hypoelliptic, 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.
