Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager-Machlup approach

TL;DR

This paper addresses the complexities of defining and manipulating Langevin equations with multiplicative white noise in the path integral framework, proposing an extended stochastic calculus to ensure consistency.

Contribution

It introduces a modified stochastic calculus for the path integral representation of Langevin equations with multiplicative noise, resolving inconsistencies in transformations.

Findings

Identifies the origin of inconsistencies in path-integral transformations.

Proposes an extended Itô calculus for path integrals.

Ensures consistent variable changes in the Onsager-Machlup form.

Abstract

The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known It\=o prescription ($dB^2=dt$) in a way that defines a modified stochastic calculus to be used inside the path-integral representation of the process, in its Onsager-Machlup form.