## Electronic Communications in Probability

### Transition density of a hyperbolic Bessel process

#### Abstract

We investigate the transition density of a hyperbolic Bessel process for integer dimensions and show a link between transition densities of a hyperbolic Brownian motion and a Bessel process in the same dimension. Using the so-called Millson’s formula for the densities of hyperbolic Brownian motion we also show a link between transition density of $n$-dimensional hyperbolic Bessel process and $2$-dimensional (if $n$ is even) or $3$-dimensional (if $n$ is odd) hyperbolic Brownian motion. This helps us to get explicit formulas for the Bessel process transition density.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 50, 10 pp.

Dates
Accepted: 5 July 2016
First available in Project Euclid: 26 July 2016

https://projecteuclid.org/euclid.ecp/1469557024

Digital Object Identifier
doi:10.1214/16-ECP8

Mathematical Reviews number (MathSciNet)
MR3533282

Zentralblatt MATH identifier
1345.60092

Subjects

#### Citation

Pyć, Andrzej; Żak, Tomasz. Transition density of a hyperbolic Bessel process. Electron. Commun. Probab. 21 (2016), paper no. 50, 10 pp. doi:10.1214/16-ECP8. https://projecteuclid.org/euclid.ecp/1469557024

#### References

• [1] A.N. Borodin, Hypergeometric diffusion, Journal of Mathematical Sciences 3, 295–304, (2009).
• [2] A.N. Borodin, P. Salminen, Handbook of Brownian Motion - Facts and Formulae, Second Edition, Birkhäuser, (2002).
• [3] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, London, (1984).
• [4] L. Gallardo, M. Yor, Some new examples of Markov processes which enjoy the time-inversion property, Probab. Theory Relat. Fields 132, 150–162, (2005).
• [5] A. Grigor’yan, M. Noguchi, The heat kernel on hyperbolic space, Bulletin of the London Mathematical Society 30, 643–650, (1998).
• [6] J-C. Gruet, Semi-groupe du mouvement Brownien hyperbolique, Stochastics Stochastic Rep. 56, 53–61, (1996).
• [7] J-C. Gruet, Windings of hyperbolic Brownian motion, In: Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers, Biblioteca de la Revista Matemática Iberoamericana, 35–72, (1997).
• [8] J-C. Gruet, A note on hyperbolic von Mises distributions, Bernoulli 6, 1007–1020, (2000).
• [9] J. Jakubowski, M. Wiśniewolski, On hyperbolic Bessel process and beyond, Bernoulli 19, 2437–2454, (2013).
• [10] T.H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, In: R.A. Askey, T.H. Koornwinder and W.S. Schempp (ed). Special functions, group theoretic aspects and applications, Reidel, Dordrecht, (1984).
• [11] H. Matsumoto, M. Yor, Exponential functionals of Brownian motion, II: Some related diffusion processes, Probab. Surveys 2, 348–384, (2005).
• [12] J.W. Pitman, L.C.G. Rogers, Markov functions, Ann. Probab. 9, 573–582, (1981).
• [13] M. Ryznar, T. Żak, Exit time of a hyperbolic $\alpha$-stable process from a half-space or a ball, Potential Analysis 45, 83–107, (2016).
• [14] J. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, Springer, (2006).