A Law of Large Numbers in the Supremum Norm for a Multiscale Stochastic Spatial Gene Network

Debussche Arnaud and Nguepedja Nankep Mac Jugal ()
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Debussche Arnaud: Institut de Recherche Mathematique de Rennes, Rennes, Bretagne, France
Nguepedja Nankep Mac Jugal: Institut de Recherche Mathematique de Rennes, Rennes, Bretagne, France

The International Journal of Biostatistics, 2019, vol. 15, issue 2, 39

Abstract: We study the asymptotic behavior of multiscale stochastic spatial gene networks. Multiscaling takes into account the difference of abundance between molecules, and captures the dynamic of rare species at a mesoscopic level. We introduce an assumption of spatial correlations for reactions involving rare species and a new law of large numbers is obtained. According to the scales, the whole system splits into two parts with different but coupled dynamics. The high scale component converges to the usual spatial model which is the solution of a partial differential equation, whereas the low scale component converges to the usual homogeneous model which is the solution of an ordinary differential equation. Comparisons are made in the supremum norm.

Keywords: probabilities; stochastic processes; biochemical modeling; Epidemiology (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1515/ijb-2017-0091

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