Isolated singularities in the heat equation behaving like fractional Brownian motions

TL;DR

This paper investigates isolated singularities in solutions to the heat equation, showing conditions under which they are removable, especially when the singularity's motion resembles fractional Brownian motion with a critical Hurst exponent.

Contribution

It establishes the removability criteria for singularities depending on their strength and the H"older continuity of their motion, including the case of fractional Brownian motion.

Findings

Singularities are removable if weaker than a certain order related to the H"older exponent.

Existence of solutions with nonremovable singularities demonstrates the optimality of the criteria.

The critical Hurst exponent for fractional Brownian motion is identified as H=1/N.

Abstract

We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is H\"older continuous with the exponent $\alpha \in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $\alpha$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2] $. It turns out that $H=1/N$ is critical.