Supports of Certain Infinitely Divisible Probability Measures on Locally Convex Spaces

Abstract

Let $\mathbf{B}$ be a separable Banach space and let $\mu$ be a centered Poisson probability measure on $\mathbf{B}$ with Levy measure $M$. Assume that $M$ admits a polar decomposition in terms of a finite measure $\sigma$ on the unit sphere of $\mathbf{B}$ and a Levy measure $\rho$ on $(0, \infty)$. The main result of this paper provides a complete description of the structure of $\mathscr{J}_\mu$, the support of $\mu$. Specifically, it is shown that: (i) if $\int_{(0, 1\lbrack} s\rho(ds) = \infty$, then $\mathscr{J}_\mu$ is a linear space and is equal to the closure of the semigroup generated by $\mathscr{J}_M$ (the support of $M$) and the negative of the barycenter of $\sigma$; and (ii) if $\int_{(0, 1\rbrack} s\rho(ds) < \infty$ and zero is in the support of $\rho$, then $\mathscr{J}_\mu$ is a convex cone and is equal to the closure of the semigroup generated by $\mathscr{J}_M$. The result (i) yields an affirmative answer to the question, open for some time, of whether the support of a stable probability measure of index $1 \leq \alpha < 2$ on $B$ is a translate of a linear space. Analogs of these results, for both Poisson and stable probability measures defined on general locally convex spaces, are also provided.