Ornstein-Uhlenbeck approximation of one-step processes: a differential equation approach

TL;DR

This paper presents a differential equation approach to approximate the steady state of one-step processes using Fokker-Planck equations, providing analytic formulas with proven accuracy of order $1/N^{eta}$.

Contribution

It introduces a novel PDE-based method to approximate steady states of density-dependent one-step processes with proven accuracy bounds.

Findings

Analytic formulas for steady state distributions are derived.

The approximation accuracy is proven to be of order $1/N^{eta}$.

The method is heuristic but rigorously justified.

Abstract

The steady state of the Fokker-Planck equation corresponding to a density dependent one-step process is approximated by a suitable normal distribution. Starting from the master equations of the process, written in terms of the time dependent probabilities, $p_k(t)$ of the states $k=0,1,\ldots , N$, their continuous (in space) version, the Fokker-Planck equation is formulated. This PDE approximation enables us to create analytic approximation formulas for the steady state distribution. These formulas are derived based on heuristic reasoning and then their accuracy is proved to be of order $1/N^{\beta}$ with some power $\beta <1$.