Intertwining and commutation relations for birth-death processes

Given a birth-death process on $\mathbb{N}$ with semigroup $(P_t{)}_{t\ge 0}$ and a discrete gradient ${\mathrm{\partial }}_u$ depending on a positive weight $u$, we establish intertwining relations of the form ${\mathrm{\partial }}_u{P}_t=Q_t{\mathrm{\partial }}_u$, where $(Q_t{)}_{t\ge 0}$ 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.