Symmetric Ornstein-Uhlenbeck Semigroups and their Generators

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dc.creatorChojnowska-Michalik, A.
dc.creatorGoldys, B.
dc.date2002-05-30
dc.date.accessioned2026-07-07T04:48:47Z
dc.date.available2026-07-07T04:48:47Z
dc.descriptionWe provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $μ$. For a reversible process the domain of its generator in $L^p(μ)$ is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup $(R_t)$ to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of $R_t$ and provide an estimate for the (appropriately defined) gradient of $R_tϕ$. Finally, in the Hilbert-Schmidt case, we show that for any $ϕ\in L^p(μ)$ the function $R_tϕ$ is an (almost) classical solution of a version of the Kolmogorov equation.
dc.identifierhttps://arxiv.org/abs/math/0205315
dc.identifierhttp://arxiv.org/abs/math/0205315
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/64186
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60H15, 47F05
dc.titleSymmetric Ornstein-Uhlenbeck Semigroups and their Generators
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