Generalized dimensions of images of measures under Gaussian processes

TL;DR

This paper establishes a formula for the generalized dimensions of measures transformed by certain Gaussian processes, linking the dimensions to properties of the processes like Holder continuity and local nondeterminism.

Contribution

It provides a new theoretical result connecting the generalized dimensions of measures with the properties of Gaussian processes, including fractional Brownian motion and related fields.

Findings

Derived a formula for D_q(mu_X) involving process parameters

Applicable to fractional Brownian motion and Riesz-Bessel motions

Shows almost sure equality of dimensions under specified conditions

Abstract

We show that for certain Gaussian random processes and fields X:R^N to R^d, D_q(mu_X) = min{d, D_q(mu)/alpha} a.s. for an index alpha which depends on Holder properties and strong local nondeterminism of X, where q>1, where D_q denotes generalized q-dimension and where mu_X is the image of the measure mu under X. In particular this holds for index-alpha fractional Brownian motion, for fractional Riesz-Bessel motions and for certain infinity scale fractional Brownian motions.