The Annals of Applied Probability

Convergence of stochastic gene networks to hybrid piecewise deterministic processes

A. Crudu, A. Debussche, A. Muller, and O. Radulescu

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 5 (2012), 1822-1859.

Dates
First available in Project Euclid: 12 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1350067987

Digital Object Identifier
doi:10.1214/11-AAP814

Mathematical Reviews number (MathSciNet)
MR3025682

Zentralblatt MATH identifier
1261.60073

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 92B05: General biology and biomathematics

Keywords
Stochastic gene networks piecewise deterministic processes perturbed test functions

Citation

Crudu, A.; Debussche, A.; Muller, A.; Radulescu, O. Convergence of stochastic gene networks to hybrid piecewise deterministic processes. Ann. Appl. Probab. 22 (2012), no. 5, 1822--1859. doi:10.1214/11-AAP814. https://projecteuclid.org/euclid.aoap/1350067987


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