From almost sure local regularity to almost sure Hausdorff dimension for Gaussian fields

Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France
erick.herbin@ecp.fr; benjamin.arras@ecp.fr; geoffroy.barruel@student.ecp.fr

Received: 5 June 2012
Revised: 6 February 2013

Abstract

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball B(t₀,ρ) are bounded from above using the local Hölder exponent at t₀. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the sample path on an open interval are controlled almost surely by the minimum of the local exponents. Then, we apply these generic results to the cases of the set-indexed fractional Brownian motion on R^N, the multifractional Brownian motion whose regularity function H is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.