Year : 2013, Volume : 4, Issue : 1
First page : ( 1) Last page : ( 18)
Print ISSN : 0973-4317. Online ISSN : 1945-919X. Published online : 2013  1.

On Random Dynamical Systems and Levels of Their Description

Illner Reinhard¹, Chevrotière Michèle De La*

¹Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4

*Supported by a grant from the Natural Sciences and Engineering Research Council of Canada; AMS Classification 92C37, 60K20, 60K35

Abstract

We consider dynamical systems in which a (typically vector-valued) dependent variable evolves according to autonomous dynamics switching randomly according to Markovian laws that change with the value of the dependent variable. Such systems are known as “random evolutions” or, in electrical engineering contexts, as “switching systems”. Systems of this type are encountered in applications from electrical engineering to cell biology (our paper was inspired by a recent model for genetic oscillators). We review the derivation of the forward Kolmogorov master equations for the probabilities to find the system in a certain state at some time. In the limit of an infinite switching rate solutions of our system converge almost surely to solutions of an averaged problem. The classical tool of a Chapman-Enskog expansion, well-known from kinetic theory, provides diffusion approximations to first order in the scale parameter. Our work focusses on a few typical examples. The analysis is formal, but we expect that the results hold rigorously in analogy to earlier results from kinetic theory (see [11]).

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