# 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.

## Key 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.