## 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 C

_{0}-semigroup on a Banach space E and W_{H}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 language | English |
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Title of host publication | Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics |

Editors | Wolfgang Arendt, Ralph Chill, Yuri Tomilov |

Place of Publication | Cham |

Publisher | Birkhäuser |

Pages | 353-368 |

Number of pages | 16 |

ISBN (Electronic) | 978-3-319-18494-4 |

ISBN (Print) | 978-3-319-18493-7 |

DOIs | |

Publication status | Published - 11 Dec 2015 |

### Publication series

Name | Operator Theory: Advances and Applications |
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Volume | 250 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 2296-4878 |

## Keywords

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

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