Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Palle Jorgensen, Jay Jorgenson, Lejla Smajlovic

TL;DR

The paper tackles explicit construction of heat kernels and Green's functions in measure spaces by starting from a parametrized small-time approximation (a parametrix) and building the true heat kernel as a convergent Neumann series. It develops a unified framework for L^p spaces, Hilbert spaces, and graphs, introducing generalized time convolution to merge spatial and temporal components, and proving convergence, uniqueness, and semigroup properties. Key contributions include explicit Neumann-series heat-kernel representations in $L^1$, $L^2$, and Hilbert-space settings, parametrix constructions across three settings, and connections to Green's functions, resistance metrics, and entropy, with concrete examples on graphs and reproducing-kernel Hilbert spaces. The results provide a flexible, robust toolkit for diffusion-type problems on general measure spaces, enabling explicit kernels and derived objects in broad contexts and suggesting directions for further study in non-separable spaces and metric changes.

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.

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

TL;DR

The paper tackles explicit construction of heat kernels and Green's functions in measure spaces by starting from a parametrized small-time approximation (a parametrix) and building the true heat kernel as a convergent Neumann series. It develops a unified framework for L^p spaces, Hilbert spaces, and graphs, introducing generalized time convolution to merge spatial and temporal components, and proving convergence, uniqueness, and semigroup properties. Key contributions include explicit Neumann-series heat-kernel representations in , , and Hilbert-space settings, parametrix constructions across three settings, and connections to Green's functions, resistance metrics, and entropy, with concrete examples on graphs and reproducing-kernel Hilbert spaces. The results provide a flexible, robust toolkit for diffusion-type problems on general measure spaces, enabling explicit kernels and derived objects in broad contexts and suggesting directions for further study in non-separable spaces and metric changes.

Abstract

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as , , and Hilbert spaces, associated to a locally compact Hausdorff space with Borel -algebra , 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.
Paper Structure (38 sections, 18 theorems, 183 equations)

This paper contains 38 sections, 18 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Imagining the heat kernel in the physical world
  3. A general point of view
  4. Related studies
  5. Organization of the paper.
  6. Our setup
  7. Transfer operators and conductance
  8. The Laplacian
  9. Hilbert spaces and measure spaces of interest
  10. Properties of the Laplacians
  11. Heat kernel on $L^{p}$ spaces and Hilbert spaces
  12. Generalized time convolution
  13. Generalized time convolution on measurable spaces
  14. Properties of the generalized time convolution on a Hilbert space
  15. Generalized time convolution on a Hilbert space
  16. ...and 23 more sections

Key Result

Lemma 2.3

Let $f:\mathfrak{X}\times(0,\infty)$ be a function such that: (i) for any $t\in(0,\infty)$, function $x\mapsto f(x,t)$ belongs to $\mathcal{H}$, and (ii) for any $x\in\mathfrak{X}$, function $t\mapsto f(x,t)$ is differentiable. Further assume that Then

Theorems & Definitions (61)

  • Remark 1
  • Remark 2
  • Definition 2.1
  • Remark 3
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Definition 3.1
  • Definition 3.2
  • ...and 51 more