Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities

TL;DR

This paper establishes upper and lower bounds for the heat kernel of a differential operator with a singular drift in Euclidean space, using isoperimetric inequalities related to a specific measure.

Contribution

It introduces new heat kernel bounds for an operator with a singular drift, derived from novel isoperimetric inequalities for a weighted measure.

Findings

Derived explicit heat kernel bounds for the operator

Established isoperimetric inequalities for a weighted measure

Connected isoperimetric results to heat kernel estimates

Abstract

We prove upper and lower bounds of the heat kernel for the operator $\Delta-\nabla (\frac{1}{|x|^{\alpha}})\cdot \nabla $ in $\mathbb{R}^{n}\setminus\{0} $ where $\alpha >0$. We obtain these bounds from an isoperimetric inequality for a measure $\mathrm{e}^{-\frac{1}{|x|^{\alpha}}}dx$ on $\mathbb{R}^{n}\setminus \{0\}$. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.