@inbook{60b1c60d385545a0aaa358c8640a4529,

title = "The Lp-Poincar{\'e} Inequality for Analytic Ornstein–Uhlenbeck Semigroups",

abstract = "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 Poincar{\'e} 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 Fr{\'e}chet derivative in the direction of H.",

keywords = "Analytic Ornstein–Uhlenbeck semigroups, Poincar{\'e} inequality, compact resolvent , joint functional calculus",

author = "{van Neerven}, Jan",

year = "2015",

month = dec,

day = "11",

doi = "10.1007/978-3-319-18494-4_23",

language = "English",

isbn = "978-3-319-18493-7",

series = "Operator Theory: Advances and Applications",

publisher = "Birkh{\"a}user",

pages = "353--368",

editor = "Wolfgang Arendt and Ralph Chill and Yuri Tomilov",

booktitle = "Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics",

}