Solving partial differential equation via stochastic process

Abstract

We investigate a theoretical framework to solve partial differential equations by using stochastic processes, e.g., chemical reaction systems. The framework is based on 'duality' concept in stochastic processes, which has been widely studied and used in mathematics and physics. Using the duality concept, a partial differential equation is connected to a stochastic process via a duality function. Without solving the partial differential equation, information about the partial differential equation can be obtained from a solution of the stochastic process. From a viewpoint of unconventional computation, one may say that the stochastic process can solve the partial differential equation. An algebraic method to derive dual processes is explained, and two examples of partial differential equations are shown. © Springer-Verlag Berlin Heidelberg 2010.