Extended duality relations between birth-death processes and partial differential equations

TL;DR

This paper introduces extended duality relations linking diffusion-type partial differential equations to discrete-state stochastic processes, utilizing the Doi-Peliti formalism and algebraic probability theory to broaden the scope of duality applications.

Contribution

It presents a novel extension of duality relations that allows deriving discrete stochastic processes from arbitrary diffusion PDEs, incorporating additional states as needed.

Findings

Extended duality relations derived using Doi-Peliti formalism.

Additional states are sometimes necessary for the discrete processes.

Framework broadens the applicability of duality in stochastic modeling.

Abstract

Duality relations between continuous-state and discrete-state stochastic processes with continuous-time have already been studied and used in various research fields. We propose extended duality relations, which enable us to derive discrete-state stochastic processes from arbitrary diffusion-type partial differential equations. The derivation is based on the Doi-Peliti formalism and the algebraic probability theory, and it will be clarified that additional states for the discrete-state stochastic processes must be considered in some cases.