Model reduction methods for classical stochastic systems with fast-switching environments: reduced master equations, stochastic differential equations, and applications

TL;DR

This paper develops reduced models for classical stochastic systems with fast-switching environments, deriving master equations and stochastic differential equations, and applies these methods to biological and physical systems.

Contribution

It introduces new reduction techniques for stochastic systems with fast environmental switching, including handling of negative rates and bursting events, with applications to biology and physics.

Findings

Derived reduced master equations with corrections for fast-switching environments

Developed a simulation algorithm for unraveling complex master equations

Applied methods to biological systems and crack propagation models

Abstract

We study classical stochastic systems with discrete states, coupled to switching external environments. For fast environmental processes we derive reduced dynamics for the system itself, focusing on corrections to the adiabatic limit of infinite time scale separation. In some cases, this leads to master equations with negative transition `rates' or bursting events. We devise a simulation algorithm in discrete time to unravel these master equations into sample paths, and provide an interpretation of bursting events. Focusing on stochastic population dynamics coupled to external environments, we discuss a series of approximation schemes combining expansions in the inverse switching rate of the environment, and a Kramers--Moyal expansion in the inverse size of the population. This places the different approximations in relation to existing work on piecewise-deterministic and piecewise-diffusive Markov processes. We apply the model reduction methods to different examples including systems in biology and a model of crack propagation.