Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups

Abstract

Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$Lf(x)=\frac{1}{2}\mathrm{Tr}Q{D}^{2}f(x)+<Ax,Df(x)>.$$ Here $Q$ is a positive symmetric operator from $E^{\ast }$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under the assumption that $P$ admits an invariant measure $\mu$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then $$\n P(t)-P(s)\n_{L^1(E,μ)} = 2$$ for all $t,s\ge 0$ 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 $A\ne 0$ there exists $t_0>0$ such that $$\n P(t)-P(s)\n_{BUC(E)} = 2$$ for all $0\le t,s\le t_0$ with $t\ne s$. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either $$ \n P(t)- P(s)\n_{BUC(E)} = 2$$ for all $t,s\ge 0$, \ $t\ne s$, 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 $L^1(E,\mu )$ and $BUC(E)$.

14 pages; to appear in J. Evolution Equations