A covariant fermionic path integral for scalar Langevin processes with multiplicative white noise

TL;DR

This paper develops a covariant fermionic path integral framework for scalar Langevin processes with multiplicative white noise, ensuring invariance under variable transformations and connecting to Onsager-Machlup formulations.

Contribution

It introduces a covariant fermionic path integral approach for multiplicative noise Langevin processes, directly formulated in continuous time, and clarifies transformation properties of auxiliary variables.

Findings

Derived a covariant fermionic path integral for Langevin processes.

Ensured covariance under non-linear variable transformations.

Connected the fermionic formulation to Onsager-Machlup in continuous time.

Abstract

We revisit the construction of the fermionic path-integral representation of overdamped scalar Langevin processes with multiplicative white noise, focusing on the covariance of the generating functional under non-linear changes of variables. We identify the transformations of the auxiliary (commuting and anticommuting) variables that ensure covariance under such transformations. The subtleties induced by the non-differentiable trajectories of the stochastic dynamics are encoded in the fermionic statistics. Upon integrating out the auxiliary variables, we derive the Onsager-Machlup formulation, which agrees with the one recently obtained using a higher-order discretization scheme. In contrast to the latter, the construction proposed here is formulated directly in continuous time.