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On the uniqueness for the heat equation with density on infinite graphs

Giulia Meglioli

TL;DR

This work addresses the uniqueness of solutions to the parabolic equation with density, $\rho\partial_t u-\Delta u=0$, on infinite weighted graphs. By constructing energy-type a priori estimates with carefully chosen test functions that reflect the geometry and the density behavior, the authors identify two distinct uniqueness regimes: one where the density is bounded below and another where the density may decay at infinity. In each regime, they prove that the only solution in a specified summability class is the trivial one, extending prior Euclidean and manifold results to graphs and highlighting how density decay and intrinsic graph geometry determine uniqueness. The results advance understanding of parabolic Cauchy problems on graphs and have implications for stochastic completeness-like properties in discrete settings.

Abstract

We study the uniqueness of solutions to a class of heat equations with positive density posed on infinite weighted graphs. We separately consider the case when the density is bounded from below by a positive constant and the case of possibly vanishing density, showing that these two scenarios lead to two different classes of uniqueness.

On the uniqueness for the heat equation with density on infinite graphs

TL;DR

This work addresses the uniqueness of solutions to the parabolic equation with density, , on infinite weighted graphs. By constructing energy-type a priori estimates with carefully chosen test functions that reflect the geometry and the density behavior, the authors identify two distinct uniqueness regimes: one where the density is bounded below and another where the density may decay at infinity. In each regime, they prove that the only solution in a specified summability class is the trivial one, extending prior Euclidean and manifold results to graphs and highlighting how density decay and intrinsic graph geometry determine uniqueness. The results advance understanding of parabolic Cauchy problems on graphs and have implications for stochastic completeness-like properties in discrete settings.

Abstract

We study the uniqueness of solutions to a class of heat equations with positive density posed on infinite weighted graphs. We separately consider the case when the density is bounded from below by a positive constant and the case of possibly vanishing density, showing that these two scenarios lead to two different classes of uniqueness.
Paper Structure (15 sections, 14 theorems, 168 equations)

This paper contains 15 sections, 14 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Outline of our results
  3. Organization of the paper
  4. Mathematical framework
  5. The graph setting
  6. Difference and Laplace operators on graphs
  7. Statement of the main results
  8. Density bounded away from zero
  9. Vanishing density
  10. Auxiliary results on infinite graphs
  11. Some useful functions and their properties
  12. Density bounded away from zero
  13. Vanishing density
  14. Proof of Theorems \ref{['teo2']} and \ref{['teo1']}
  15. Proof of Theorems \ref{['teo3']} and \ref{['teo4']}

Key Result

Lemma 2.4

Let $\psi\in C^1(\mathbb R; \mathbb R)$ be a convex function, let $u\in \mathfrak F$. Then

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5: Laplacian of the product
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • ...and 24 more