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Norm discontinuity and spectral properties of Ornstein–Uhlenbeck semigroups. (English) Zbl 1117.47035

Let \(BUC(E)\) the space of bounded real-valued uniformly continuous functions on a real Banach space \(E\). The authors consider an Ornstein–Uhlenbeck semigroup \(P=\{P(t)\}_{t\geq 0}\) defined on \(BUC(E)\) such that \(P\) admits an invariant measure. They investigate the condition \[ \|P(t)-P(s)\| = 2, \quad s,t\geq 0,\;s\neq t, \tag{*} \] where the norm is taken in \({\mathcal L}(BUC(E))\), the Banach space of all bounded linear operators on \(BUC(E)\). They prove that \((*)\) holds, under some regularity assumptions. Moreover, the behaviour of \(P\) in \(BUC(E)\) and a dichotomy related to \((*)\) are studied. Some results of the authors are new even for finite-dimensional spaces. As an application, they investigate the spectrum of the generator associated to \(P\).

MSC:

47D07 Markov semigroups and applications to diffusion processes
35J70 Degenerate elliptic equations
35P05 General topics in linear spectral theory for PDEs
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
60J35 Transition functions, generators and resolvents
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