Stable laws arising from hitting distributions of processes on homogeneous trees and the hyperbolic half-plane

Paolo Baldi, Enrico Casadio Tarabusi and Alessandro Figà-Talamanca

Vol. 197 (2001), No. 2, 257–273

DOI: 10.2140/pjm.2001.197.257

Abstract

The projective line with respect to a local field is the boundary of the Bruhat-Tits tree associated to the field, much in the same way as the real projective line is the boundary of the upper half-plane. In both cases we may consider the horocycles with respect to the point at infinity. These horocycles are exactly the horizontal lines {y = a} with a > 0 in the real case, while in the case of a local field the horocycles may be thought of as discretizations of the field obtained by collapsing to a point each ball of a given radius.

In this paper we exploit this geometric parallelism to construct symmetric α-stable random variables on the real line and on a local field by completely analogous procedures. In the case of a local field the main ingredient is a drifted random walk on the tree. In the real case the random walk is replaced by a drifted Brownian motion on the hyperbolic half-plane. In both cases the random processes are invariant under the automorphisms of the tree and the hyperbolic half-plane, respectively, that fix the point at infinity.

These random processes determine hitting distributions on the horocycles which, in a sense to be specified, are shown to be in the domain of attraction of α-stable symmetric random variables. In both cases the exponent of α-stability is related by an explicit formula to the drift coefficient.

Milestones

Received: 5 May 1999

Published: 1 February 2001