Asymptotic Properties of Semigroups of Measures on Vector Spaces

Abstract

Let $(E, B)$ be a measurable vector space and $q$ be a measurable seminorm on $E$. Suppose that $(\mu_t)_{t > 0}$ is a $q$-continuous convolution semigroup of probability measures on $(E, B)$. It is proved that there exists a right-continuous nonincreasing function $\theta$ such that $\lim_{t \rightarrow 0+} (1/t)\cdot \mu_t\{x: q(x) > s\} = \theta(s)$ for every $s > 0$ at which $\theta$ is continuous. If $\mu_t, t > 0$, are Gaussian, then $\theta \equiv 0$; if there exists a measurable linear functional $f$ such that $f(\cdot)$ is not Gaussian (with respect to $\mu_1$) and $q \geqslant |f|$ then $\theta \not\equiv 0$.