Feynman-Kac theory of time-integrated functionals: It\^o versus functional calculus

TL;DR

This paper develops a unified Feynman-Kac framework for analyzing time-integrated functionals in stochastic processes using Itô and functional calculus, providing new derivations and connecting different approaches in the field.

Contribution

It introduces a novel derivation of the Feynman-Kac theory via functional calculus and unifies various methods used in stochastic thermodynamics.

Findings

Derived Feynman-Kac theory using Itô calculus and functional calculus.

Recapitulated recent results on steady-state (co)variances of dynamical functionals.

Provided more accessible derivations to connect different approaches in the literature.

Abstract

The fluctuations of dynamical functionals such as the empirical density and current as well as heat, work and generalized currents in stochastic thermodynamics are usually studied within the Feynman-Kac tilting formalism, which in the Physics literature is typically derived by some form of Kramers-Moyal expansion, or in the Mathematical literature via the Cameron-Martin-Girsanov approach. Here we derive the Feynman-Kac theory for general additive dynamical functionals directly via It\^o calculus and via functional calculus, where the latter result in fact appears to be new. Using Dyson series we then independently recapitulate recent results on steady-state (co)variances of general additive dynamical functionals derived recently in Dieball and Godec ({2022 \textit{Phys. Rev. Lett.}~\textbf{129} 140601}) and Dieball and Godec ({2022 \textit{Phys. Rev. Res.}~\textbf{4} 033243}). We hope for our work to put the different approaches to the statistics of dynamical functionals employed in the field on a common footing, and to illustrate more easily accessible ways to the tilting formalism.