Abstract
We study the stochastic processes that are images of Brownian motions on Heisenberg group H 2n+1 under conformal maps. In particular, we obtain that Cayley transform maps Brownian paths in H 2n+1 to a time changed Brownian motion on CR sphere \(\mathbb{S}^{2n+1}\) conditioned to be at its south pole at a random time. We also obtain that the inversion of Brownian motion on H 2n+1 started from x ≠ 0, is up to time change, a Brownian bridge on H 2n+1 conditioned to be at the origin.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
F. Baudoin, M. Bonnefont, The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds. Math. Z. 263, 647–672 (2009)
F. Baudoin, J. Wang, The subelliptic heat kernel on the CR sphere. Math. Z. 275(1–2), 135–150 (2013)
F. Baudoin, J. Wang, Stochastic areas, winding numbers and Hopf fibrations. Probab. Theory Relat. Fields (2016). https://doi.org/10.1007/s00440-016-0745-x
R. Beals, B. Gaveau, P.C. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. 79(7), 633–689 (2000)
T.K. Carne, Brownian motion and stereographic projection. Ann. Inst. Henri Poincaré Probab. Stat. Sect. B 21(2), 187–196 (1985)
S. Dragomir, G. Tomassini, Differential Geometry and Analysis on CR Manifolds, vol. 246 (Birkhäuser, Boston, 2006)
B. Gaveau, Principe de moindre action, propagation de la chaleur et estiméees sous elliptiques sur certains groupes nilpotents. Acta Math. 139(1), 95–153 (1977)
D. Geller, The Laplacian and the Kohn Laplacian for the sphere. J. Differ. Geom. 15, 417–435 (1980)
A. Korányi, Kelvin transforms and harmonic polynomials on the Heisenberg group. J. Funct. Anal. 49(2), 177–185 (1982)
L. Schwartz, Le mouvement brownien sur \(\mathbb{R}^{N}\), en tant que semi-martingale dans S N . Ann. Inst. Henri Poincaré Probab. Stat. 21(1), 15–25 (1985)
M. Yor, A prtopos de l’inverse du mouvement brownien dans \(\mathbb{R}^{n}\,(n \geq 3)\). Ann. Inst. Henri Poincaré Probab. Stat. 21(1), 27–38 (1985)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Wang, J. (2017). Conformal Transforms and Doob’s h-Processes on Heisenberg Groups. In: Baudoin, F., Peterson, J. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol 72. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59671-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-59671-6_8
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-59670-9
Online ISBN: 978-3-319-59671-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)