Intertwining and commutation relations for birth-death processes

TL;DR

This paper establishes intertwining relations for birth-death processes using discrete gradients, leading to new insights into functional inequalities, contraction properties, and stochastic orderings, with simple proofs based on interpolation and convexity.

Contribution

It introduces a novel intertwining relation involving Feynman-Kac semigroups for birth-death processes, expanding the theoretical framework and applications.

Findings

Lipschitz contraction and Wasserstein curvature results

Functional inequalities derived from the intertwining relations

Stochastic orderings established for birth-death processes

Abstract

Given a birth-death process on $\mathbb {N}$ with semigroup $(P_t)_{t\geq0}$ and a discrete gradient ${\partial}_u$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_uP_t=Q_t\,{\partial}_u$, where $(Q_t)_{t\geq0}$ is the Feynman-Kac semigroup with potential $V_u$ of another birth-death process. We provide applications when $V_u$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.