Harmonic analysis of additive Levy processes

TL;DR

This paper investigates the conditions under which the Minkowski sum of the range of additive Levy processes and a set has positive Lebesgue measure, linking it to capacity and potential densities, and extends previous results.

Contribution

It removes symmetry conditions from earlier work and provides new characterizations of the range's measure properties using harmonic analysis and potential densities.

Findings

Range plus set has positive Lebesgue measure iff a certain capacity is positive.

Necessary and sufficient conditions expressed via one-potential densities.

Develops harmonic analysis results with implications for Levy process theory.

Abstract

Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional L\'evy processes, and consider the $N$-parameter random field \[\X(\bm{t}):= X_1(t_1)+...+X_N(t_N).\] First we demonstrate that for all nonrandom Borel sets $F\subseteq\R^d$, the Minkowski sum $\X(\R^N_+)\oplus F$, of the range $\X(\R^N_+)$ of $\X$ with $F$, can have positive $d$-dimensional Lebesgue measure if and only if a certain capacity of $F$ is positive. This improves our earlier joint effort with Yuquan Zhong \ycite{KXZ:03} by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in \fullocite{KXZ:03}, the potential theory of the type studied here has a large number of consequences in the theory of L\'evy processes. We present a few new consequences here.