Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings

TL;DR

This paper investigates the long-term behavior of heat equation solutions on affine buildings, introducing p-mass functions to describe asymptotics and extending classical results to non-Archimedean and exotic settings.

Contribution

It introduces p-mass functions for affine buildings and extends asymptotic heat kernel analysis beyond classical Euclidean and symmetric spaces.

Findings

Caloric functions asymptotically factor into mass functions and heat kernels.

Characterization of heat kernel concentration in p-norms.

Identification of different asymptotic regimes based on p-values.

Abstract

We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with initial data belonging to certain weighted-$\ell^1$ spaces or to the radial $\ell^1$ class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat--Tits framework. We characterize the spatial concentration of heat kernels in $p$-norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of $p$, and clarify the interplay between volume growth and heat diffusion.