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arXiv · 2512.24348

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

Palle Jorgensen·Jay Jorgenson·Lejla Smajlovic
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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 $σ$-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.

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Palle Jorgensen, Jay Jorgenson, Lejla Smajlovic. 2025-12-30. An explicit construction of heat kernels and Green's functions in measure spaces. https://arxiv.org/abs/2512.24348

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