Feynman-Kac theory of time-integrated functionals: Itô versus functional calculus

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.