Polarity of points for Gaussian random fields in critical dimension

TL;DR

This paper investigates when points are hit or avoided by Gaussian random fields in critical dimensions, providing new integral criteria and extending existing methods to establish polarity conditions.

Contribution

It introduces an optimal integral criterion for polarity of points in Gaussian fields with specific variance functions, extending Talagrand's covering argument.

Findings

Points are polar if a certain integral diverges in the critical dimension.

The integral criterion is both necessary and sufficient for the specific case.

Extension of Talagrand's covering argument to the critical dimension.

Abstract

We study the property of hitting points for a class of $\mathbb{R}^d$-valued continuous Gaussian random fields on $\mathbb{R}^N$ with stationary increments, i.i.d. coordinates, and a regularly varying variance function $\sigma$ of index $0<H<1$. We first prove that if \[ \lim_{r\to 0^+} \frac{r^N}{\sigma^d\left(r\left( \log\log\frac{1}{r}\right)^{-1/N}\right)} = \infty, \] then every fixed point is polar (i.e., not hit almost surely). In general, this criterion may not be optimal in the critical dimension $d=N/H$. To aim for an optimal condition, we consider the specific case $\sigma(r) = r^H (\log(1/r))^\gamma$ and prove that, in the critical dimension $d=N/H$, points are polar if and only if $\gamma \le 1/d$, or equivalently in this specific case, \[ \int_{0^+} \frac{r^{N-1}}{\sigma^d(r)} dr = \infty. \] This integral condition is also necessary for points to be polar under general assumptions. Our main contribution lies in the proof of sufficiency of this condition in the specific case, where we extend a covering argument of Talagrand (1998) based on sojourn time estimates to obtain Hausdorff measure bounds and solve polarity of points in the critical dimension.