The chemical birth-death process with additive noise

TL;DR

This paper explores eight exact methods to solve the chemical birth-death process with additive noise, providing insights into stochastic chemical kinetics through analytical solutions and approximations.

Contribution

It systematically compares eight different exact and approximate techniques for solving the stochastic chemical birth-death process, highlighting their applications and differences.

Findings

Eight qualitatively different exact solution methods demonstrated.

Analytical solutions for stochastic differential equations and path integrals provided.

Insights into stochastic effects on chemical kinetics obtained.

Abstract

The chemical birth-death process, whose chemical master equation (CME) is exactly solvable, is a paradigmatic toy problem often used to get intuition for how stochasticity affects chemical kinetics. In a certain limit, it can be approximated by an Ornstein-Uhlenbeck-like process which is also exactly solvable. In this paper, we use this system to showcase eight qualitatively different ways to exactly solve continuous stochastic systems: (i) integrating the stochastic differential equation; (ii) computing the characteristic function; (iii) eigenfunction expansion; (iv) using ladder operators; (v) the Martin-Siggia-Rose-Janssen-De Dominicis path integral; (vi) the Onsager-Machlup path integral; (vii) semiclassically approximating the Onsager-Machlup path integral; and (viii) approximating the solution to the corresponding CME.