# Feynman-Kac Equations for Reaction and Diffusion Processes

## TL;DR

This paper develops a comprehensive theoretical framework for deriving Feynman-Kac equations that describe the distribution of functionals of particles undergoing diffusion and chemical reactions, applicable to various diffusion types and reaction rates.

## Contribution

It introduces general forms of forward and backward Feynman-Kac equations for reaction-diffusion processes, including normal and anomalous diffusion with linear and nonlinear reactions.

## Key findings

- Derived equations for different diffusion and reaction types
- Analyzed occupation time and first passage time using the equations
- Included boundary conditions like absorbing and reflecting

## Abstract

This paper provides a theoretical framework of deriving the forward and backward Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing both diffusion and chemical reaction. Very general forms of the equations are obtained. Once given the diffusion type and reaction rate, a specific forward or backward Feynman-Kac equation can be obtained. The listed in the paper include the ones for normal/anomalous diffusions and reactions with linear/nonlinear rates. Using the derived equations, we also study the occupation time in half-space, the first passage time to a fixed boundary, and the occupation time in half-space with absorbing or reflecting boundary conditions.