arXiv subjects

Shmuel Rakotonirina-Ricquebourg

Publications and source records attributed to Shmuel Rakotonirina-Ricquebourg.

3 recordsLinked to original sources

arXiv2021

Diffusion limit for a stochastic kinetic problem with unbounded driving process

This paper studies the limit of a kinetic evolution equation involving a small parameter and driven by a random process which also scales with the small parameter. In order to prove the convergence in distribution to the solution of a stochastic diffusion equation while removing a boundedness assumption on the driving random process, we adapt the method of perturbed test functions to work with stopped martingales problems.

math.PR↗

arXiv2021

Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters $ε$ and $δ$: the evolution equation contains singular terms with respect to $ε$, and is driven by a fast ergodic process which evolves at the time scale $t/δ^2$. We prove that when $(ε,δ)\to (0,0)$ the density converges to the solution of a linear diffusion PDE. This is a mixture of diffusion approximation in the PDE sense (with respect to the parameter $ε$) and of averaging in the probabilistic sense (with respect to the parameter $δ$). The proof employs stopping times arguments and a suitable perturbed test functions approach which is adapted to consider the general regime $ε\neq δ$.

math.PR↗

arXiv2020

On Asymptotic Preserving schemes for a class of Stochastic Differential Equations in averaging and diffusion approximation regimes

We introduce and study a notion of Asymptotic Preserving schemes, related to convergence in distribution, for a class of slow-fast Stochastic Differential Equations. In some examples, crude schemes fail to capture the correct limiting equation resulting from averaging and diffusion approximation procedures. We propose examples of Asymptotic Preserving schemes: when the time-scale separation vanishes, one obtains a limiting scheme, which is shown to be consistent in distribution with the limiting Stochastic Differential Equation. Numerical experiments illustrate the importance of the proposed Asymptotic Preserving schemes for several examples. In addition, in the averaging regime, error estimates are obtained and the proposed scheme is proved to be uniformly accurate.

math.NA↗