Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups

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 $$Lf(x) = \frac{1}{2}{\text{Tr}}\,QD^2 f(x) + \left\langle {Ax,Df(x)} \right\rangle ,\quad x \in E.$$ 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 $$\left\| {P(t) - P(s)} \right\|_{\fancyscript{L} (L^1 (E,\mu _\infty ))} = 2\quad {\text{for all }}t,\,s\,\geq\,0{\text{ with }}t\,\ne\,s.$$ 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 $$\left\| {P(t) - P(s)} \right\|_{\fancyscript{L}(BUC(E))} = 2\quad {\text{for all }}0\,\leq\,t,\,s\,\leq\,t_0 {\text{ with }}t\,\ne\,s.$$ Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either $$\left\| {P(t) - P(s)} \right\|_{\fancyscript{L}(BUC(E))} = 2\quad {\text{for all }}t,\,s\,\geq\,0,\;t\,\ne\,s,$$ 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).