## Local properties of Lévy processes on a totally disconnected group.(English)Zbl 0683.60010

The author investigates the path behavior of Lévy processes $$X=(X(t):$$ $$t\in {\mathbb{R}}_+)$$ on a probability space ($$\Omega$$,$${\mathcal M},P)$$ taking values in a non-discrete metrizable, totally disconnected Abelian locally compact group G. Such processes correspond to continuous convolution semigroups $$(\mu_ t:$$ $$t\in {\mathbb{R}}_+)$$ of probability measures on G which admit Lévy-Khintchine representations with Lévy measures $$\nu$$ on $$G\setminus \{0\}$$ describing the jumps of X. In fact, if G is a Vilenkin group defined by a sequence $$G_ 0\supset G_ 1\supset G_ 2\supset...\supset \{0\}$$ of compact open subgroups, then $$\nu (G\setminus G_ n)<\infty$$ for all $$n\geq 0.$$
At first the sets of times at which X is fast or slow for a non- decreasing function $$h:\quad {\mathbb{R}}_+\to {\mathbb{R}}_+$$ such that $$\lim_{t\to 0}h(t)=0$$, are studied in terms of the measure $$\nu$$ (Theorems 2 and 3). Next, the growth of the first exit times and sojourn times of X are discussed (Theorem 4). Various versions of the variation of the paths of X are shown to exist, to coincide within the general framework and to be representable in terms of the jumps of X (Theorem 6). Finally, the author applies the notions of Hausdorff- and packing dimensions to measuring the image X([0,t]) of a compact interval [0,t] under the process X (Theorems 8 and 10). These “fractal” aspects of the theory are then extended to sets other than intervals yielding uniform dimension results (Theorems 11 and 12).
A typical example concerns the group G of p-adic integers: If $$\nu$$ is locally spherically symmetric and $$\nu (G\setminus G_ n)=p^{\alpha n}$$ for some $$\alpha\in [0,1]$$ and $$r\geq 0$$, then $$P(\dim \quad X(B)=\alpha \dim B\quad for\quad all\quad B\subset {\mathbb{R}}_+)=1.$$ The main results of the paper under review have been inspired by their Euclidean analogues.
Reviewer: H.Heyer

### MSC:

 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60G17 Sample path properties 60J99 Markov processes
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### References:

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