Random fractals and Markov processes

Abstract

This is a survey on the sample path properties of Markov processes, especially fractal properties of the random sets and measures determined by their sample paths. The class of Markov processes considered in this paper include Lévy processes in R d , diffusions on fractals and on R d , Feller processes determined by pseudo-differential operators and so on. We summarize recent results for Lévy processes such as the Hausdorff and packing dimensions of their ranges, level sets, and multiple points; regularity properties of local times and self-intersection local times; multifractal analysis of the occupation measures and sample paths. Our emphasis is on general Markovian techniques that are applicable to other Markov processes.

Dimensional Properties of Fractional Brownian Motion

“…Since then, the problem of establishing uniform Hausdorff dimension results has been studied by several authors for various classes of stochastic processes. See Xiao [19] for a survey on the results for Markov processes and their applications. Monrad and Pitt [10] have proved the following uniform Hausdorff dimension result for the images of B α : If N ≤ αd, then almost surely…”

Additive Lévy Processes: Capacity and Hausdorff Dimension

“…Since then, this method has become one of the standard tools in obtaining lower bounds for the Hausdorff dimension of random sets. We refer to the survey papers of Fristedt (1974), Taylor (1986) and Xiao (2003) for further information about the results and techniques for Lévy processes, and extensive lists of references.…”

Sample Path Properties of Anisotropic Gaussian Random Fields

“…We mention that various tools from fractal geometry have been applied to studying sample path properties of stochastic processes since 1950's. The survey papers of Taylor (1986) and Xiao (2004) summarize various fractal properties of random sets related to sample paths of Markov processes. Throughout the rest of this paper, we will assume that…”