Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes

Abstract:

A probability measure $\mu$ on $\mathbb {R}^d$ is called weakly unimodal if there exists a constant $\kappa \ge 1$ such that for all $r>0$, \begin{equation} \sup _{a\in \mathbb {R}^d} \mu (B(a, r)) \le \kappa \mu (B(0, r)). \end{equation} Here, $B(a, r)$ denotes the $\ell ^\infty$-ball centered at $a\in \mathbb {R}^d$ with radius $r>0$. In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of $\mathbb {R}^d$. In particular, we use this to prove that every symmetric infinitely divisible distribution is weakly unimodal. This result is then applied to improve some recent results of the authors on capacities and level sets of additive Lévy processes.