Abstract.
Let E be a real Banach space. We study the Ornstein-Uhlenbeck semigroup P = {P(t)} t ≥ 0 associated with the Ornstein-Uhlenbeck operator
Here \(Q \in \fancyscript{L}(E^*,\,E)\) is a positive symmetric operator and A is the generator of a C 0-semigroup S = {S(t)} t ≥ 0 on E. Under the assumption that P admits an invariant measure μ∞ we prove that if S is eventually compact and the spectrum of its generator is nonempty, then
This result is new even when \(E = \mathbb{R}^n .\) We also study the behaviour of P in the space BUC(E). We show that if A ≠ 0 there exists t 0 > 0 such that
Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either
or S is the direct sum of a nilpotent semigroup and a finite-dimensional periodic semigroup. Finally we investigate the spectrum of L in the spaces L 1 (E, μ∞) and BUC(E).
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Neerven, J.M.A.M.v., Priola, E. Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups. J. evol. equ. 5, 557–576 (2005). https://doi.org/10.1007/s00028-005-0239-2
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DOI: https://doi.org/10.1007/s00028-005-0239-2