An explicit construction of heat kernels and Green's functions in measure spaces

TL;DR

This paper presents an explicit method to construct heat kernels and Green's functions in measure spaces, extending classical techniques to new settings like graphs and reproducing kernel Hilbert spaces.

Contribution

It introduces a Neumann series approach for constructing heat kernels in various measure-theoretic contexts, including graphs and Hilbert spaces, based on a parametrix approximation.

Findings

Constructed heat kernels for $L^{1}$, $L^{2}$, and Hilbert spaces.

Extended classical methods to graphs and reproducing kernel spaces.

Provided explicit series representations for heat kernels.

Abstract

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $\sigma$-algebra $\mathcal{B}$, and endowed with additional measure-theoretic data. Our approach is an adaptation of classical work due to Minakshishundaram and Pleijel, and it requires as input a parametrix or small time approximation to the heat kernel. The methodology developed in this article applies to yield new instances of heat kernel constructions, including normalized Laplacians on finite and infinite graphs as well as Hilbert spaces with reproducing kernels.