Packing dimension of images and graphs of Gaussian random fields with drift

TL;DR

This paper determines the almost sure packing dimension of images and graphs of Gaussian random fields with drift, extending previous results and providing new bounds especially for fractional Brownian motion.

Contribution

It generalizes existing results to include Gaussian fields with drift and offers new bounds for fractional Brownian motion graphs, even in one-dimensional cases.

Findings

Calculated the packing dimension of images and graphs of Gaussian fields with drift.

Provided sharp lower bounds for the packing dimension of fractional Brownian motion graphs.

Extended previous results to more general Gaussian random fields and functions.

Abstract

Let $X=\{(X_1(t),\dots,X_d(t)): t\in \mathbb{R}^n\}$ be a Gaussian random field in $\mathbb{R}^d$ such that $X_1,\dots,X_d$ are independent, centered Gaussian random fields with continuous sample paths. Let $f\colon \mathbb{R}^n\to \mathbb{R}^d$ be a Borel map and let $A\subset \mathbb{R}^n$ be an analytic set. The main goal of the paper is to determine the almost sure value of the packing dimension of the image and graph of $X+f$ restricted to $A$ under a very mild assumption. This generalizes a result of Du, Miao, Wu and Xiao, who calculated the packing dimension of $X(A)$ if $X_1,\dots,X_d$ are independent copies of the same Gaussian random field $X_0$. Provided that $X$ is a fractional Brownian motion, our result is new even if $n=d=1$ and $f$ is continuous, and even if $f\equiv 0$ in the case of graphs. For a fractional Brownian motion $X$ we also obtain the sharp lower bound for the packing dimension of the graph of $X$ over $A$ in terms of the Hurst index of $X$ and the packing dimension of $A$. The analogous result for images was obtained by Talagrand and Xiao.