Exact results for a noise-induced bistable system

TL;DR

This paper provides exact solutions for a noise-induced bistable system, including mean switching time and dynamics, avoiding continuous approximations used in prior studies.

Contribution

It derives exact solutions for the discrete stochastic system, extending previous work that relied on continuous Fokker-Planck approximations.

Findings

Exact expression for mean switching time

Exact solutions for moments and equilibrium time

Validation of discrete model results

Abstract

A stochastic system where bistability is caused by noise has been recently investigated by Biancalani et al. (PRL 112:038101, 2014). They have computed the mean switching time for such a system using a continuous Fokker-Planck equation derived from the Taylor expansion of the Master equation to estimate the parameter of such a system from experiment. In this article, we provide the exact solution for the full discrete system without resorting to continuous approximation and obtain the expression for the mean switching time. We further extend this investigation by solving exactly the Master equation and obtaining the expression of other quantities of interests such as the dynamics of the moments and the equilibrium time.