Asymptotic Properties of Semigroups of Measures on Vector Spaces
Asymptotic Properties of Semigroups of Measures on Vector Spaces
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: …