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hal-00534598, version 1

Intertwining and commutation relations for birth-death processes

Djalil Chafai () 1, Aldéric Joulin () 2

(2010-11-10)

Abstract: 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.

  • 1:  Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
  • Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout
  • 2:  Institut de Mathématiques de Toulouse (IMT)
  • Université Paul Sabatier [UPS] - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées (INSA) - Toulouse – CNRS : UMR5219
  • Domain : Mathematics/Probability
  • Keywords : Birth-death process – Feynman-Kac semigroup – discrete gradients – intertwining relation – functional inequalities
  • Available versions :  v1 (2010-11-10) v2 (2011-08-21) v3 (2012-03-11)
 
  • hal-00534598, version 1
  • http://hal.archives-ouvertes.fr/hal-00534598
  • oai:hal.archives-ouvertes.fr:hal-00534598
  • From: Djalil Chafaï <>
  • Submitted on: Wednesday, 10 November 2010 10:23:42
  • Updated on: Wednesday, 10 November 2010 11:06:54