Convergence of stochastic gene networks to hybrid piecewise deterministic processes

Alina Crudu (1) , Arnaud Debussche (2, 1) , Aurélie Muller (3, 4) , Ovidiu Radulescu (5)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IPSO - Invariant Preserving SOlvers
3 IECN - Institut Élie Cartan de Nancy
4 BIGS - Biology, genetics and statistics
5 Département de Mathématiques [Montpellier]

We study the asymptotic behavior of multiscale stochastic gene networks using weak limits of Markov jump processes. Depending on the time and concentration scales of the system we distinguish four types of limits: continuous piecewise deterministic processes (PDP) with switching, PDP with jumps in the continuous variables, averaged PDP, and PDP with singular switching. We justify rigorously the convergence for the four types of limits. The convergence results can be used to simplify the stochastic dynamics of gene network models arising in molecular biology.

Mots clés: Stochastic gene networks; piecewise deterministic processes; perturbed test functions

Domaines: Probabilités [math.PR]

Alina Crudu, Arnaud Debussche, Aurélie Muller, Ovidiu Radulescu. Convergence of stochastic gene networks to hybrid piecewise deterministic processes. The Annals of Applied Probability, 2012, 22 (5), pp.1822-1859. ⟨10.1214/11-AAP814⟩. ⟨hal-00553482⟩