Duality relations between spatial birth-death processes and diffusions in Hilbert space

TL;DR

This paper establishes duality relations between spatial birth-death processes and diffusions in Hilbert space using Doi field theory, enabling new analytical methods for complex stochastic models.

Contribution

It introduces a novel duality framework connecting birth-death processes and Hilbert space diffusions via Doi formalism, facilitating path integral calculations.

Findings

Duality relations enable analytical calculations of expectations.

Applications demonstrated in cable signalling and jump-diffusion models.

Framework bridges finite and infinite-dimensional stochastic processes.

Abstract

Spatially dependent birth-death processes can be modelled by kinetic models such as the BBGKY hierarchy. Diffusion in infinite dimensional systems can be modelled with Brownian motion in Hilbert space. In this work Doi field theoretic formalism is utilised to establish dualities between these classes of processes. This enables path integral methods to calculate expectations of duality functions. These are exemplified with models ranging from stochastic cable signalling to jump-diffusion processes.