Master equations and the theory of stochastic path integrals

TL;DR

This paper reviews master equations and their path integral representations, discussing analytical and numerical methods for solving them, especially under strong stochastic fluctuations, with applications to chemical reactions and rare event probabilities.

Contribution

It introduces novel mappings of master equations to PDEs and derives exact path integral representations, enhancing analytical tools for stochastic process analysis.

Findings

Path integral representations provide exact solutions to master equations.

Spectral, WKB, and variational methods are effective for PDE analysis.

Path integrals can approximate rare event probabilities.

Abstract

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approximation of rare event probabilities and derive path integral representations of Fokker-Planck equations. To make our review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.