Local properties of Lévy processes on a totally disconnected group
type: Item
instance of: scholarly article
title: Local properties of Lévy processes on a totally disconnected group
zbMATH Open document ID: 0683.60010
DOI: 10.1007/BF01053411
author: Steven Neil Evans
published in: Journal of Theoretical Probability
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 4119321
zbMATH Keywords: sample paths of processes with independent increments on totally disconnected Abelian groups
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: path behavior of Lévy processes
zbMATH Keywords: locally compact group
zbMATH Keywords: Lévy-Khintchine representations
zbMATH Keywords: sojourn times
zbMATH Keywords: packing dimensions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4119321