# Integrability of Stochastic Birth-Death processes via Differential Galois Theory

## TL;DR

This paper investigates the mathematical integrability of stochastic birth-death processes using differential Galois theory, revealing that only linear rate cases are integrable, which impacts the solvability of these models.

## Contribution

It applies differential Galois theory to analyze the integrability of birth-death processes with polynomial rates, providing new insights into their solvability.

## Key findings

- PDE is non-integrable for polynomial rates
- Linear rate cases are integrable
- Differential Galois theory effectively analyzes stochastic process integrability

## Abstract

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the (trivial) case in which rates are linear functions of the number of individuals.