Path integral representation for stochastic jump processes with boundaries

TL;DR

This paper introduces a novel path integral formalism using su(2) Lie algebra to analyze finite-state stochastic jump processes, enabling concise calculation of generating functions especially in epidemic models.

Contribution

It develops a new quantum-mechanics-inspired path integral approach for finite-state stochastic processes, extending analytical tools available for such systems.

Findings

Path integral representation for stochastic processes with boundaries.

Application to SIS epidemic model with time-dependent rates.

Efficient calculation of probability generating functions.

Abstract

We propose a formalism to analyze discrete stochastic processes with finite-state-level N. By using an (N+1)-dimensional representation of su(2) Lie algebra, we re-express the master equation to a time-evolution equation for the state vector corresponding to the probability generating function. We found that the generating function of the system can be expressed as a propagator in the spin coherent state representation. The generating function has a path integral representation in terms of the spin coherent state. We apply our formalism to a linear Susceptible-Infected-Susceptible (SIS) epidemic model with time-dependent transition probabilities. The probability generating function of the system is calculated concisely using an algebraic property of the system or a path integral representation. Our results indicate that the method of analysis developed in the field of quantum mechanics is applicable to discrete stochastic processes with finite-state-level.