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Transition semigroups for stochastic semilinear equations on Hilbert spaces

Tom 396 / 2001

Anna Chojnowska-Michalik Dissertationes Mathematicae 396 (2001), 1-59 MSC: 60H15, 47D07, 60J35, 47D06. DOI: 10.4064/dm396-0-1

Streszczenie

A large class of stochastic semilinear equations with measurable nonlinear term on a Hilbert space $H$ is considered. Assuming the corresponding nonsymmetric Ornstein–Uhlenbeck process has an invariant measure $\mu$, we prove in the $L^p(H, \mu)$ spaces the existence of a transition semigroup $(P_t)$ for the equations. Sufficient conditions are provided for hyperboundedness of $P_t$ and for the Log Sobolev Inequality to hold; and in the case of a bounded nonlinear term, sufficient and necessary conditions are obtained. We prove the existence, uniqueness and some regularity of an invariant density for $(P_t)$. A characterization of the domain of the generator is also given. The main tools are the Girsanov transform and Miyadera perturbations.

Autorzy

  • Anna Chojnowska-MichalikFaculty of Mathematics
    University of /L/od/x
    Banacha 22
    90-238 /L/od/x, Poland
    e-mail

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