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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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