Local times of anisotropic Gaussian random fields and stochastic heat equation

TL;DR

The paper develops a unified theory for the local times, Hölder regularity, and level-set geometry of anisotropic Gaussian random fields with strong local nondeterminism in an anisotropic metric $\rho(t,s)=\sum_{j=1}^N|t_j-s_j|^{H_j}$. By leveraging Voronoi partitions under $\rho$ and Besicovitch covering, it obtains sharp moment bounds for local times and their increments, leading to precise Hölder conditions and Chung's law of the iterated logarithm for sample paths. It then determines the exact Hausdorff measure gauge for level sets with respect to $\rho$, and in the parabolic metric $\delta$, gives the corresponding gauge and fractal dimension for stochastic heat equation solutions. Applications to systems of stochastic heat equations with additive Gaussian noise, including a linear transformation $v=Au$, show how these results extend to non-stationary increments and yield exact level-set measures in both anisotropic and parabolic settings. The work thus provides sharp fractal and regularity characterizations for a broad class of Gaussian fields and their stochastic-PDE manifestations.

Abstract

We study the local times of a large class of Gaussian random fields satisfying strong local nondeterminism with respect to an anisotropic metric. We establish moment estimates and Hölder conditions for the local times of the Gaussian random fields. Our key estimates rely on geometric properties of Voronoi partitions with respect to an anisotropic metric and the use of Besicovitch's covering theorem. As a consequence, we deduce sample path properties of the Gaussian random fields that are related to Chung's law of the iterated logarithm and modulus of non-differentiability. Moreover, we apply our results to systems of stochastic heat equations with additive Gaussian noise and determine the exact Hausdorff measure function with respect to the parabolic metric for the level sets of the solutions.

Figures (1)

• Figure 1: Two cases of configuration: (1) $t_1^1 \ge t_1$ (left), and (2) $t^1_1 < t_1$ (right)

Theorems & Definitions (35)

• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• proof