Work

Intertwining and commutation relations for birth–death processes

Year: 2013

Type: article

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 ${\partial}_{u}P_{t}=Q_{t}\,{\partial}_{u}$, 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.

Source: Bernoulli

DOI: 10.3150/12-bej433

Authors: Djalil Chafaï, Aldéric Joulin