Symmetric Ornstein-Uhlenbeck semigroups and their generators

• 成果类型: Article
• 署名作者: Chojnowska-Michalik, A; Goldys, B
• 署名单位: University of Lodz; University of New South Wales Sydney
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400200222
• 发表日期: 2002
• 页码: 459-486
• 关键词: logarithmic sobolev inequalities; invariant-measures; operator; SPACES; EQUIVALENCE; ergodicity

摘要

We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure mu. For a reversible process the domain of its generator in L-p(mu) 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-l)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-l and provide an estimate for the (appropriately defined) gradient of R(t)phi. Finally, in the Hilbert-Schmidt case, we show that for any phi is an element of L-p(mu) the function R(t)phi is an (almost) classical solution of a version of the Kolmogorov equation.