The Lp-Poincaré Inequality for Analytic Ornstein–Uhlenbeck Semigroups

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Consider the linear stochastic evolution equation dU(t)=AU(t)dt+dW H (t),t⩾0, dU(t)=AU(t)dt+dWH(t),t⩾0, where A generates a C0-semigroup on a Banach space E and WH is a cylindrical Brownian motion in a continuously embedded Hilbert subspace H of E. Under the assumption that the solutions to this equation admit an invariant measure µ∞ we prove that if the associated Ornstein–Uhlenbeck semigroup is analytic and has compact resolvent, then the Poincaré inequality ∥ ∥ f−f ¯ ∥ ∥  L p (E,μ ∞ ) ⩽∥D H f∥ L p (E,μ ∞ )  ‖f−f¯‖Lp(E,μ∞)⩽‖DHf‖Lp(E,μ∞) holds for all 1 < p < ∞. Here f denotes the average of f with respect to µ∞ and DH the Fréchet derivative in the direction of H.
Original languageEnglish
Title of host publicationOperator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics
EditorsWolfgang Arendt, Ralph Chill, Yuri Tomilov
Place of PublicationCham
Number of pages16
ISBN (Electronic)978-3-319-18494-4
ISBN (Print)978-3-319-18493-7
Publication statusPublished - 11 Dec 2015

Publication series

NameOperator Theory: Advances and Applications
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878


  • Analytic Ornstein–Uhlenbeck semigroups
  • Poincaré inequality
  • compact resolvent
  • joint functional calculus

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