Derivation of Stochastic Partial Differential Equations for Reaction-Diffusion Processes

Abstract

Stochastic partial differential equations are derived for the reaction-diffusion process in one, two, and three dimensions. Specifically, stochastic partial differential equations are derived for the random dynamics of particles that are reacting and diffusing in a medium. In the derivation, a discrete stochastic reaction-diffusion equation is first constructed from basic principles, that is, from the changes that occur in a small time interval. As the time interval goes to zero, the discrete stochastic model leads to a system of It stochastic differential equation. As the spatial intervals approach zero, a stochastic partial differential equation is derived for the reaction-diffusion process. The stochastic reaction-diffusion equation can be solved computationally using numerical methods for systems of It stochastic differential equations. Comparisons between numerical solutions of stochastic reaction-diffusion equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.