Feynman-Kac equation revisited

TL;DR

This paper revisits the Feynman-Kac equation, deriving new governing equations for path functionals of Langevin systems with complex force fields and noise, with applications demonstrated through occupation time and area calculations.

Contribution

The paper introduces generalized Feynman-Kac equations for Langevin systems with space-time dependent forces and multiplicative noise, expanding the theoretical framework for path functionals.

Findings

Derived governing equations for path functionals of Langevin systems.

Validated equations through simulations of occupation time and area.

Extended Feynman-Kac framework to complex stochastic dynamics.

Abstract

Functionals of particles' paths have diverse applications in physics, mathematics, hydrology, economics, and other fields. Under the framework of continuous time random walk (CTRW), the governing equations for the probability density functions (PDFs) of the functionals, including the ones of the paths of stochastic processes of normal diffusion, anomalous diffusion, and even the diffusion with reaction, have been derived. Sometimes, the stochastic processes in physics and chemistry are naturally described by Langevin equations. The Langevin picture has the advantages in studying the dynamics with an external force field and analyzing the effect of noise resulting from a fluctuating environment. We derive the governing equations of the PDFs of the functionals of paths of Langevin system with both space and time dependent force field and arbitrary multiplicative noise; and the backward version is proposed for the system with arbitrary additive noise or multiplicative Gaussian white noise together with a force field. For the newly built equations, their applications of solving the PDFs of the occupation time and area under the trajectory curve are provided, and the results are confirmed by simulations.