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

TL;DR

This paper establishes a connection between local regularity and Hausdorff dimensions for multiparameter Gaussian fields, providing bounds and applying results to fractional Brownian motions and Weierstrass functions.

Contribution

It introduces the deterministic local sub-exponent for Gaussian processes and links local H"older exponents to Hausdorff dimensions, extending understanding of sample path regularity.

Findings

Bounds on Hausdorff dimensions of Gaussian fields' range and graph

Lower bounds for dimensions using local sub-exponent

Application to fractional Brownian motion and Weierstrass functions

Abstract

Fine regularity of stochastic processes is usually measured in a local way by local H\"older exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,\rho)$ are bounded from above using the local H\"older exponent at $t_0$. 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 multiparameter fractional Brownian motion, the multifractional Brownian motion whose regularity function $H$ is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.