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Random motions at finite velocity in a non-Euclidean space

Part of: Global differential geometry Special processes

Published online by Cambridge University Press:  01 July 2016

E. Orsingher  and
A. De Gregorio
E. Orsingher*
Affiliation:
University of Rome ‘La Sapienza’
A. De Gregorio*
Affiliation:
University of Padua
*
∗ Postal address: Dipartimento di Statistica, Probabilità e Statistiche Applicate, University of Rome ‘La Sapienza’, Piazzale Aldo Moro 5, Rome, 00185, Italy. Email address: enzo.orsingher@uniroma1.it
∗∗ Current address: Dipartimento di Scienze Economiche, Aziendali e Statistiche, University of Milan, via del Conservatorio 7, Milano, 20122, Italy.
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Abstract

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In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

Keywords

Hyperbolic distancetelegraph processLobachevsky geometryhyperbolic equationgeodesic lineBougerol's identity

MSC classification

Primary: 60K40: Other physical applications of random processes
Secondary: 53C22: Geodesics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 588 - 611
Copyright
Copyright © Applied Probability Trust 2007 

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