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Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees

Effie Papageorgiou

Abstract

We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\to\infty$. Second, using them, we show that solutions with initial data in weighted $\ell^1$ classes, asymptotically factorize in $\ell^p$ norms, $p\in[1,\infty]$, as the product of the heat kernel, times a $p$-mass function, dependent on the initial condition and $p$. The $p$-mass function is described in terms of boundary averages associated with Busemann functions for $p<2$, while for $p\ge 2$, it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all $p$, emphasizing the influence of the graph geometry on heat diffusion.

Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees

Abstract

We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as . Second, using them, we show that solutions with initial data in weighted classes, asymptotically factorize in norms, , as the product of the heat kernel, times a -mass function, dependent on the initial condition and . The -mass function is described in terms of boundary averages associated with Busemann functions for , while for , it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all , emphasizing the influence of the graph geometry on heat diffusion.
Paper Structure (19 sections, 16 theorems, 202 equations)

This paper contains 19 sections, 16 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Homogeneous trees
  3. Boundary, Poisson Kernel and Harmonic Analysis
  4. The spherical Fourier transform on homogeneous trees
  5. The Helgason-Fourier transform
  6. The heat kernel on homogeneous trees
  7. Heat kernel asymptotics
  8. Asymptotics for ratios of heat kernels
  9. Long time behavior of solutions to the heat equation on the integers
  10. Long time behavior of solutions to the heat equation on a homogeneous tree
  11. Finitely supported initial data.
  12. The case $1\leq p< 2$.
  13. Outside the critical $\ell^p$ region
  14. Inside the critical region.
  15. The case $2<p\leq \infty$
  16. ...and 4 more sections

Key Result

Theorem A

Let $h_t$ denote the heat kernel on a homogeneous tree and $h_t^{\mathbb{Z}}$ that of the integers. Then, for $x\in \mathscr{T}$ such that $|x|\to \infty$ and $t\to \infty$, we have the following asymptotics: where $c$ is an explicit constant depending on the rate of $|x|/t$. Instead, if $x\in \mathscr{T}$ is such that $|x|\leq \rho(t)$, with $\frac{\rho(t)}{\sqrt{t}}\rightarrow 0$ as $t\rightarr

Theorems & Definitions (30)

  • Theorem A
  • Theorem B
  • Theorem 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 20 more