Level Sets of Additive Lévy Processes

Weak Unimodality of Finite Measures, and an Application to Potential Theory of Additive Lévy Processes

A probability measure μ on Rd is called weakly unimodal if there exists a constant κ ≥ 1 such that for all r > 0, (0.1) sup a∈Rd μ(B(a, r)) ≤ κμ(B(0, r)). Here, B(a, r) denotes the `∞-ball centered at a ∈ Rd with radius r > 0. In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of Rd. In particular, we use this to prove that every symmetric infi...

Additive Lévy Processes

Moderate Deviations and Laws of the Iterated Logarithm for the Local times of Additive Lévy Processes and Additive Random Walks

We study the upper tail behaviors of the local times of the additive Lévy processes and additive random walks. The limit forms we establish are the moderate deviations and the laws of the iterated logarithm for the L2-norms of the local times and for the local times at a fixed site. Subject classifications: 60F10, 60F15, 60J55, 60G52

Additive Lévy Processes: Capacity and Hausdorff Dimension

This is a survey on recently-developed potential theory of additive Lévy processes and its applications to fractal geometry of Lévy processes. Additive Lévy processes arise naturally in the studies of the Brownian sheet, intersections of Lévy processes and so on. We first summarize some recent results on the novel connections between an additive Lévy process X in R , and a natural class of ener...

Additive functionals of several Lévy processes and self-intersection local times

Different extentons of an isomorphism theorem of Dynkin are developed and are used to study two distinct but related families of functionals of Lévy processes; n-fold “near-intersections” of a single Lévy process, which is also referred to as a self-intersection local time, and continuous additive functionals of several independent Lévy processes. Intersection local times for n independent Lévy...