Sample function properties of multi-parameter stable processes

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

“…First we mention the following papers which are closely related to the topics of this paper, but will not be further addressed because all the random fields considered there possess certain Markovian nature. Ehm (1981) has established many deep results on the sample path properties of the stable sheet and his arguments rely crucially on the property of independent increments of the stable sheet. Khoshnevisan, Xiao and Zhong (2003a, b) have extended several of Ehm's results to additive Lévy processes and have also established some useful connections between hitting probabilities and a class of natural capacities.…”

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

“…In the special case of the Brownian sheet, the Hausdorff measure of the range and graph were evaluated by Ehm [13]. However, his method relies crucially on the independent increment property of the Brownian sheet and is not applicable to the fractional Brownian sheet B H in general.…”

Sample Path Properties of Anisotropic Gaussian Random Fields

“…For example, it would be interesting to determine the exact Hausdorff and packing measure functions for the range X [0, 1] N and graph GrX [0, 1] N for anisotropic Gaussian random fields. When X is the Brownian sheet or a fractional Brownian motion, the problems on exact Hausdorff measure functions have been considered by Ehm (1981), Talagrand (1995Talagrand ( , 1998, Xiao (1996aXiao ( , 1997aXiao ( , 1997b. Here is a summary of the known results: …”

“…As such, they are useful in studying the various fractal properties of level sets and inverse images of the vector field X. In this regard, we refer to Berman (1972), Ehm (1981), Rosen (1984), Monrad and Pitt (1987) and Xiao (1997b). First we consider the existence of the local times of Gaussian random fields.…”

“…Proof The proofs of (8.51) and (8.52) are based on Lemma 8.12 and a chaining argument, which is the same as those of Theorems 1.1 and 1.2 in Xiao (1997b) [see also Ehm (1981)]. We leave the details to the reader.…”