Mass functions and asymptotic behavior of caloric functions on affine buildings

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 $$p \in [1, \infty ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , we introduce a notion of a p -mass function and prove that caloric functions with initial data belonging to certain weighted- $$\ell ^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> spaces or to the radial $$\ell ^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> 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.