Gaussian lower bound for the Neumann Green function of ageneral parabolic operator

TL;DR

This paper establishes Gaussian two-sided bounds for the Neumann Green function of a general parabolic operator using perturbation and parametrix methods, with implications for heat kernels on manifolds.

Contribution

It provides a novel approach to derive Gaussian bounds for the Neumann Green function and extends the method to Riemannian manifolds with boundary.

Findings

Gaussian two-sided bounds for Neumann Green function

Simple proof for bounds of fundamental solution

Extension to heat kernels on manifolds with boundary

Abstract

Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish gaussian two-sided bounds for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for the gaussian two-sided bounds for the fundamental solution. We also indicate how our method can be adapted to get a gaussian lower bound for the Neumann heat kernel of a compact Riemannian manifold with boundary having non negative Ricci curvature.