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

摘要

A probability measure mu on R-d is called weakly unimodal if there exists a constant kappa greater than or equal to 1 such that for all r > 0, (0.1) [GRAPHICS] Here, B(a, r) denotes the l(infinity)-ball centered at a is an element of 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 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 Levy processes. [References: 11]