Symmetric path integrals for stochastic equations with multiplicative noise

TL;DR

This paper derives the correct path integral formulation for Langevin equations with multiplicative noise under a time-symmetric discretization scheme, clarifying misconceptions about the Stratonovich interpretation.

Contribution

It provides the first derivation of the path integral for multiplicative noise Langevin equations using a natural, time-symmetric discretization scheme, correcting previous assumptions.

Findings

Correct path integral formulation for multiplicative noise

Disproves the quick conversion method for Stratonovich equations

Highlights importance of discretization conventions in stochastic calculus

Abstract

A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt = - F(q) + e(q) xi, where e(q) xi is Gaussian white noise whose amplitude e(q) depends on q itself. I show how to convert such equations into path integrals. The definition of the path integral depends crucially on the convention used for discretizing time, and I specifically derive the correct path integral when the convention used is the natural, time-symmetric one that time derivatives are (q_t - q_{t-\Delta t}) / \Delta t and coordinates are (q_t + q_{t-\Delta t}) / 2. [This is the convention that permits standard manipulations of calculus on the action, like naive integration by parts.] It has sometimes been assumed in the literature that a Stratanovich Langevin equation can be quickly converted to a path integral by treating time as continuous but using the rule \theta(t=0) = 1/2. I show that this prescription fails when the amplitude e(q) is q-dependent.