Randomly switching evolution equations

TL;DR

This paper studies stochastic evolution equations driven by Markov processes, deriving moment equations and analyzing long-term behavior with biological examples illustrating bifurcations.

Contribution

It introduces a framework for analyzing PDMPs in $L^1$ and derives equations for moments and correlations, providing insights into their asymptotic behavior.

Findings

Derived equations for moments and correlations of PDMPs.

Analyzed the mean behavior at large times.

Illustrated results with biological bifurcation examples.

Abstract

We present an investigation of stochastic evolution in which a family of evolution equations in $L^1$ are driven by continuous-time Markov processes. These are examples of so-called piecewise deterministic Markov processes (PDMP's) on the space of integrable functions. We derive equations for the first moment and correlations (of any order) of such processes. We also introduce the mean of the process at large time and describe its behaviour. The results are illustrated by some simple, yet generic, biological examples characterized by different one-parameter types of bifurcations.