Intertwining and commutation relations for birth-death processes
Résumé
Given a birth-death process on N with semigroup P_t and a discrete gradient D depending on a positive weight u, we establish intertwining relations of the form D P_t = Q_t D, 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. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.
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