Section: New Results

Convergence of stochastic gene networks to hybrid piecewise deterministic processes

Participant : Arnaud Debussche.

In [47] 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 step-size. 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.