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

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.

💡 Research Summary

The paper revisits the construction of a fermionic (Grassmann) path‑integral representation for overdamped scalar Langevin equations driven by multiplicative white noise, with the aim of preserving covariance under arbitrary nonlinear changes of variables directly in continuous time. Starting from the stochastic differential equation (\dot x(t)=f(x)+g(x),\eta(t)) interpreted in the generalized α‑prescription (which includes Itô, Stratonovich and kinetic conventions), the authors first review how a nonlinear transformation (u=U(x)) modifies the drift and diffusion functions. Using the generalized chain rule they derive explicit expressions for the transformed drift (F(u)) and diffusion (G(u)) that contain only the parameter α, confirming that the higher‑order β‑term used in previous discretization schemes does not affect the continuous‑time Langevin dynamics.

The core of the work lies in Section 3, where the generating functional (Z