Harmonic analysis of additive Levy processes

Let $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_N$ denote $N$ independent $d$-dimensional L'evy processes, and consider the $N$-parameter random field $$\text{X}(\mathit{t}):=X_1(t_1)+\mathrm{.}\mathrm{.}\mathrm{.}+X_N(t_N)\mathrm{.}$$ First we demonstrate that for all nonrandom Borel sets $F\subseteq {\mathbb{R}}^d$, the Minkowski sum $\text{X}({\mathbb{R}}_{+}^N)\oplus F$, of the range $\text{X}({\mathbb{R}}_{+}^N)$ of $\text{X}$ 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.