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

TL;DR

This paper analyzes the long-time behavior of the heat kernel and solutions to the heat equation on homogeneous trees, deriving sharp asymptotics and describing how solutions factorize based on initial data and graph geometry.

Contribution

It provides new sharp asymptotic formulas for the heat kernel on homogeneous trees and characterizes the asymptotic factorization of solutions in various $ extit{ extbf{ ext{l}}}^p$ norms, highlighting the influence of tree geometry.

Findings

Sharp asymptotic formulas for the heat kernel as t→∞.

Solutions factorize into heat kernel times a p-mass function.

The p-mass function depends on boundary averages or convolution with spherical functions.

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.