ESAIM: Probability and Statistics

Special Issue: Spring School Mons Random differential equations and Gaussian fields

Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres , ∗∗

S. Cohena1 and M. A. Lifshitsa2

a1 Universitéde Toulouse, Université Paul Sabatier, Institut de Mathématiques de Toulouse, 31062 Toulouse, France.

a2 Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl., 2, 198504, Stary Peterhof, Russia;


We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.

(Received July 6 2009)

(Online publication July 3 2012)

Key Words:

  • Hyperbolic space;
  • Random fields;
  • Lévy’s Brownian field.

Mathematics Subject Classification:

  • 60G15;
  • 60G10;
  • 51M10


  ANR GDSA and grants NSh-638.2008.1, RFBR 09 − 01 − 12180-ofim.

∗∗  Most of this work was done when M. Lifshits was invited professor in Université Paul Sabatier.