Polarity of points for Gaussian random fields in critical dimension

Abstract: We study the property of hitting points for a class of $\mathbb{R}<sup>d$-valued continuous Gaussian random fields on $\mathbb{R}<sup>N$ with stationary increments, i.i.d. coordinates, and a regularly varying variance function $σ$ of index $0<H<1$. We first prove that if [ \lim_{r\to 0^+} \frac{r^N}{σ^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 $σ(r) = r^H (\log(1/r))<sup>γ$ and prove that, in the critical dimension $d=N/H$, points are polar if and only if $γ\le 1/d$, or equivalently in this specific case, [ \int_{0^+} \frac{r^{N-1}}{σ^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.

• The paper establishes a necessary and sufficient condition for point polarity in critical dimension Gaussian fields using a refined integral test.
• It employs advanced sojourn-time estimates, strong local nondeterminism, and Talagrand-style coverings to derive sharp Hausdorff measure bounds.
• The results offer precise thresholds for log-corrected variograms, linking the nonexistence of local times with point polarity in Gaussian fields.

Polarity of Points for Gaussian Random Fields in Critical Dimension

Introduction and Context

The paper "Polarity of points for Gaussian random fields in critical dimension" (2604.08129) rigorously investigates the problem of characterizing the polarity of points for a class of vector-valued, continuous Gaussian random fields with stationary increments, i.i.d. coordinates, and regularly varying variogram functions. The question of whether singletons (points) are polar—that is, whether they are almost surely not hit by the field—lies at the heart of probabilistic potential theory for Gaussian random fields and connects to fundamental aspects of hitting probabilities, the existence of local times, and critical dimension phenomena.

While earlier results, notably for the Brownian sheet and fractional Brownian motion, established polarity dichotomies hinging on dimensional relationships, the present work significantly advances the optimal description of point polarity in the critical regime where the analytic criteria become subtle and standard covering or local time arguments fail to provide sharp results. By introducing refined sojourn-time-based coverings and sharp Hausdorff-measure estimates via extensions of Talagrand’s method, this work resolves the sharp phase boundary for polarity in broad classes of Gaussian random fields, including cases where the variogram includes iterated logarithmic corrections.

Main Results and Technical Contributions

The main analytic setting is an $\mathbb{R}^d$-valued, continuous, centered Gaussian random field $X(t)$ on $\mathbb{R}^N$, with stationary increments, i.i.d. coordinates, and canonical distance $d(s,t)$ controlled above and below by a regularly varying function, $\sigma(r) = r^H L(r)$ ($L$ slowly varying, $H\in(0,1)$). The authors analyze when a fixed point $z\in\mathbb{R}^d$ is polar, i.e., whether $\mathbb{P}(\exists t\neq 0: X(t) = z) = 0$.

A sufficient condition based on the decay rate of the small ball probabilities is provided:

• If $$\lim_{r\to 0^+} \frac{r^N}{\sigma^d(r(\log\log(1/r))^{-1/N})} = \infty$$, then every point is polar almost surely.

The optimality of this condition is then examined for the critical dimension case ($X(t)$0). For the model variogram $X(t)$1, it is shown:

• In the critical regime $X(t)$2, points are polar if and only if $X(t)$3, or equivalently when

$X(t)$4

The integral divergence criterion (which can fail precisely at the critical dimension with logarithmic corrections) is shown to be necessary in general, and sufficient in the log-corrected case. Sharp Hausdorff gauge function estimates and nontrivial sojourn time lower-tail probability bounds are developed to implement a Talagrand-style covering strategy.

A main technical advance is the use of fine-grained sojourn time moment and tail bounds to obtain, in the critical dimension, for the range $X(t)$5 over compact intervals $X(t)$6,

• Almost sure finiteness of the Hausdorff $X(t)$7-measure with gauge

$X(t)$8

which implies $X(t)$9, from which polarity follows by Fubini's theorem combined with independence decomposition arguments.

Methodology Overview

The proofs blend tools from Gaussian process theory, regular variation analysis, metric entropy, and probabilistic potential theory:

• Regular Variation/Spectral Methods: The admissible Gaussian fields are characterized using Yaglom’s spectral measure representation, connecting the variogram $\mathbb{R}^N$0 with the spectral asymptotics.
• Strong Local Nondeterminism (SLND): Essential for sojourn time estimates, SLND is assumed and verified for a wide class of Gaussian random fields, ensuring quantitative independence in increments on small scales.
• Sharp Sojourn Time Estimates: High-moment and lower-tail bounds (via Paley-Zygmund) are established for the Lebesgue measure of the set $\mathbb{R}^N$1, both upper and lower, in terms of the appropriate log-corrected powers.
• Covering Arguments: The authors design a covering of the range using balls centered at $\mathbb{R}^N$2 values, with radii captured via iterated logarithmic corrections. The necessary sojourn mass for such a ball to "qualify" arises from the lower-tail estimates.
• Hausdorff Measure Bounds: The gauge function is chosen to saturate the sojourn time lower bounds, yielding exact necessary and sufficient criteria for polarity.
• Independence Decomposition and Fubini’s Theorem: Conditioning arguments and independence of increments are used to transfer measure statements from random image sets to probability of fixed-point hitting.
• Integral Tests: The equivalence between polarity and divergence of the above integral is proved, with a close correspondence to the existence of local times.

Numerical and Theoretical Implications

A major strength is the complete characterization for the log-corrected fractional case, with the following sharp transition:

• If the log exponent $\mathbb{R}^N$3 (including the equality case), points are polar; for $\mathbb{R}^N$4, they are not.
• This matches, in essence, the integral test for the existence of square-integrable local times (cf. Geman-Horowitz and Pitt), extending and making sharp previous dimension-based dichotomy results.

Additionally, the paper establishes a rigorous equivalence between the non-existence of local times and point polarity for large classes of Gaussian random fields under mild regularity and SLND conditions.

Broader Impact and Future Directions

The results provide optimal polarity criteria (and corresponding Hausdorff dimensional information) not only for fractional Brownian field models, but also for more general models where the main regularity is encoded in a variogram with regularly (and iterated logarithmically) varying behavior.

The connection between polarity and the nonexistence of local times, extended here to log-corrected critical dimension scenarios, suggests further applications in the study of SPDEs, potential theory for non-Markovian random fields, and in the modeling of multi-parameter random surfaces with critical hitting regimes.

There remains potential to extend these methods to:

• Non-Gaussian or anisotropic fields where increments may have heavy tails or different scaling in different directions.
• Dynamical models where regularity may only hold up to certain scales.
• The investigation of polarity for more general, fractal, or random target sets.

Conclusion

This work solves the sharp polarity criterion for points in a broad class of continuous, stationary increment Gaussian random fields at and above the critical dimension, including log-corrected cases where the boundary is subtle. The core contribution is a necessary and sufficient, computable integral test for polarity, fully describing the transition across the critical regime in terms of the variogram's regularity, and realizing a powerful extension of Talagrand’s sojourn-based method. The theoretical implications for the structure of random field images and the closely tied question of the existence of local times are substantial, providing conclusive answers long sought in the geometry of Gaussian fields (2604.08129).