Fine structure of the complex hyperbolic Brownian motion and Rudin’s question

• Autorzy: Graczyk, P. , Żak, T.
• Języki publikacji: EN

Abstrakty

EN

We investigate the fine structure of the complex hyperbolic Brownian motion in the unit ball of Cⁿ. It turns out that the generator of the process is locally very close to the generator of some simple transformation of the classical Brownian motion. This fact helps us to give an intuitive explanation why the invariant Laplace operator in the unit ball of Cⁿ is a difference of two ordinary Laplace operators – the question set byW. Rudin in his monograph Function Theory in the Unit Ball of Cⁿ. In the second part of the paper we find stochastic differential equations for the complex hyperbolic Brownian motion on the ball model of the complex hyperbolic space and furnish in this way an important tool in a further investigation of this process.

Słowa kluczowe

EN

complex hyperbolic space; complex hyperbolic Brownian motion; invariant Laplace operator

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2008
• Tom: Vol. 28, Fasc. 2
• Strony: 345--357
• Opis fizyczny: Bibliogr.4 poz.

Twórcy

autor

Graczyk, P.

• Département de Mathématiques, Université d’Angers, 2 boulevard Lavoisier, 49045 Angers, France,

autor

Żak, T.

• Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland,

Bibliografia

• [1] Y. Kannai, Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. Partial Differential Equations 2 (8) (1977), pp. 781-830.
• [2] H. Matsumoto, Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one, Bull. Sci. Math. 125 (2001), pp. 553-581.
• [3] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, third edition, Springer, 1999.
• [4] W. Rudin, Function Theory in the Unit Ball of Cn, Springer, 1980.