Path Integrals and Perturbation Theory for Stochastic Processes

TL;DR

This paper reviews and extends a path-integral formalism for Markov processes, demonstrating exact solutions and perturbation methods, and deriving a field theory for directed percolation from coupled stochastic processes.

Contribution

It introduces an extended path-integral approach for stochastic processes, enabling exact solutions and perturbation analysis, and connects discrete models to continuum field theories.

Findings

Exact probability generating functions for simple processes.

Diagrammatic perturbation theory for non-solvable processes.

Derivation of a field theory for directed percolation.

Abstract

We review and extend the formalism introduced by Peliti, that maps a Markov process to a path-integral representation. After developing the mapping, we apply it to some illustrative examples: the simple decay process, the birth-and-death process, and the Malthus-Verhulst process. In the first two cases we show how to obtain the exact probability generating function using the path integral. We show how to implement a diagrammatic perturbation theory for processes that do not admit an exact solution. Analysis of a set of coupled Malthus-Verhulst processes on a lattice leads, in the continuum limit, to a field theory for directed percolation and allied models.