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Diffusion on homogeneous ultrametric spaces: the contributions of Alessandro Figà-Talamanca

Wolfgang Woess

TL;DR

This work surveys diffusion on homogeneous ultrametric spaces by relating diffusion processes on boundaries of trees to discrete-time random walks on the underlying tree. It synthesizes Figà-Talamanca's notable papers (FT-41, FT-43, FT-46, FT-47) with the broader ultrametric diffusion literature, detailing hierarchical and spherically symmetric Laplacians, Taibleson-type operators on local fields such as $\\mathbb{Q}_p^n$, and the transition from random walks to convolution semigroups via Lévy measures. A central theme is the identification of boundary processes with corresponding semigroups through spherical analysis and the Naïm kernel, yielding a unified framework for diffusion on ultrametric spaces and totally disconnected groups. The notes clarify how boundary diffusion, spectral data, and harmonic analysis interlock, providing a common picture that connects discrete tree dynamics to continuous diffusion on ultrametric boundaries and local fields, with implications for understanding diffusion-type operators in $p$-adic and related settings.

Abstract

Alessandro Figà-Talamanca (1938-2023) was an influential Italian mathematician, scientific leader of the Italian group of harmonic analysis for many years. Since the late 1970ies, his interest focussed on harmonic analysis on free groups and trees. In the later years of his scientific work he became also interested in diffusion processes on homogeneous ultrametric spaces such as local fields and totally disconnected Abelian groups. This is related with the close connection of those spaces with trees and their boundaries and concerns, in particular, the construction of such processes via discrete-time walks on trees. The present notes provide rather detailed comments on this part of his work and the related, quite abundant literature. This is intended to become part of a volume of selected papers by Figà-Talamanca, accompanied by comments such as the present text.

Diffusion on homogeneous ultrametric spaces: the contributions of Alessandro Figà-Talamanca

TL;DR

This work surveys diffusion on homogeneous ultrametric spaces by relating diffusion processes on boundaries of trees to discrete-time random walks on the underlying tree. It synthesizes Figà-Talamanca's notable papers (FT-41, FT-43, FT-46, FT-47) with the broader ultrametric diffusion literature, detailing hierarchical and spherically symmetric Laplacians, Taibleson-type operators on local fields such as , and the transition from random walks to convolution semigroups via Lévy measures. A central theme is the identification of boundary processes with corresponding semigroups through spherical analysis and the Naïm kernel, yielding a unified framework for diffusion on ultrametric spaces and totally disconnected groups. The notes clarify how boundary diffusion, spectral data, and harmonic analysis interlock, providing a common picture that connects discrete tree dynamics to continuous diffusion on ultrametric boundaries and local fields, with implications for understanding diffusion-type operators in -adic and related settings.

Abstract

Alessandro Figà-Talamanca (1938-2023) was an influential Italian mathematician, scientific leader of the Italian group of harmonic analysis for many years. Since the late 1970ies, his interest focussed on harmonic analysis on free groups and trees. In the later years of his scientific work he became also interested in diffusion processes on homogeneous ultrametric spaces such as local fields and totally disconnected Abelian groups. This is related with the close connection of those spaces with trees and their boundaries and concerns, in particular, the construction of such processes via discrete-time walks on trees. The present notes provide rather detailed comments on this part of his work and the related, quite abundant literature. This is intended to become part of a volume of selected papers by Figà-Talamanca, accompanied by comments such as the present text.
Paper Structure (4 sections, 10 theorems, 56 equations, 1 figure)

This paper contains 4 sections, 10 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ultrametric spaces and trees
  3. Laplacians on ultrametric spaces
  4. Random walks on trees and processes on their boundaries

Key Result

Theorem 3.1

The operator $\mathfrak{L}$ of eq:lap is densely defined on $L^2(\mathcal{X},\mathsf{m})$ and symmetric, and admits a complete system of linearly independent, compactly supported eigenfunctions. The latter consists of all functions $f_{B,D}\,$, $D \in \mathcal{B}$ with predecessor ball $D'=B$, given The associated eigenvalue is $\lambda(B)$. In addition, when $\mathsf{m}(\mathcal{X}) < \infty\,$,

Figures (1)

  • Figure :

Theorems & Definitions (10)

  • Theorem 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.2
  • Proposition 4.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 4.1
  • Theorem 4.4