Wiener-type Criterion for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences

TL;DR

This paper establishes a Wiener-type criterion for determining when the fundamental singularity of the heat equation can be removed, linking measure-theoretic, probabilistic, and geometric conditions for singularity removability.

Contribution

It introduces a new Wiener-type criterion for the removability of singularities in the heat equation, extending previous results to arbitrary open sets and geometric boundary conditions.

Findings

Provides a geometric characterization of singularity removability.

Connects measure-theoretic and probabilistic conditions with geometric thinness.

Extends criteria to cases with locally graph-represented boundaries.

Abstract

We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context, the criterion determines whether the $h$-parabolic measure of the singularity point is null or positive. From the probabilistic point of view, the criterion presents an asymptotic law for conditional Brownian motion. In {\it U.G. Abdulla, J Math Phys, 65, 121503 (2024)} the Kolmogorov-Petrovsky-type test was established. Here, we prove a new Wiener-type criterion for the "geometric" characterization of the removability of the fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of the complementary set near the singularity point. In the special case when the boundary of the open set is locally represented by a graph, the minimal thinness criterion for the removability of the singularity is expressed in terms of the minimal regularity of the boundary manifold at the singularity point.