Intertwining and commutation relations for birth-death processes

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