Diffusion on homogeneous ultrametric spaces: the contributions of Alessandro Fig\`a-Talamanca

TL;DR

This paper discusses diffusion processes on homogeneous ultrametric spaces, highlighting Alessandro Figà-Talamanca's contributions to harmonic analysis on such spaces, especially in relation to trees and their boundaries.

Contribution

It provides detailed commentary on Figà-Talamanca's work on diffusion processes on ultrametric spaces and their connection to harmonic analysis and tree structures.

Findings

Analysis of diffusion processes on ultrametric spaces.

Connection between ultrametric spaces and trees.

Historical overview of Figà-Talamanca's contributions.

Abstract

Alessandro Fig\`a-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 Fig\`a-Talamanca, accompanied by comments such as the present text.