Norm Discontinuity and Spectral Properties of Ornstein-uhlenbeck Semigroups

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) = 1 2 TrQDf(x) + 〈Ax,Df(x)〉, x ∈ E. Here Q ∈ L (E, E) is a positive symmetric operator and A is the generator of a C0-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 ‖P (t) − P (s)‖ L (L(E,μ∞)) = 2 for all t, s ≥ 0 with t 6= s. This result is new even when E = R. We also study the behaviour of P in the space BUC(E). We show that if A 6= 0 there exists t0 > 0 such that ‖P (t) − P (s)‖L (BUC(E)) = 2 for all 0 ≤ t, s ≤ t0 with t 6= s. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either ‖P (t) − P (s)‖L (BUC(E)) = 2 for all t, s ≥ 0, t 6= 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(E,μ∞) and BUC(E).