The Annals of Probability

Isotropic Lévy processes on Riemannian manifolds

D. Applebaum and A. Estrade

Full-text: Open access


Under a natural invariance assumption on the Lévy measure we construct compound Poisson processes and more general isotropic Lévy processes on Riemannian manifolds by projection of a suitable horizontal process in the bundle of orthonormal frames.We characterize such Lévy processes through their infinitesimal generators and we show that they can be realized as the limit of a sequence of Brownian motions which are interlaced with jumps along geodesic segments.

Article information

Ann. Probab., Volume 28, Number 1 (2000), 166-184.

First available in Project Euclid: 18 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58G32 60J25: Continuous-time Markov processes on general state spaces 60E07: Infinitely divisible distributions; stable distributions
Secondary: 58G35 60G55: Point processes

Riemannian manifolds orthonormal frame bundle basic vector fields geodesics horizontal Lévy process isotropic Lévy process Feller semigroup interlacing construction


Applebaum, D.; Estrade, A. Isotropic Lévy processes on Riemannian manifolds. Ann. Probab. 28 (2000), no. 1, 166--184. doi:10.1214/aop/1019160116.

Export citation


  • [1] Applebaum,D. (1995). A horizontal L´evy process on the bundle of orthonormal frames over a complete Riemannian manifold. S´eminaire de Probabilit´es XXIX. Lecture Notes in Math. 1613 166-181. Springer, Berlin.
  • [2] Applebaum,D. (2000). Compound Poisson processes and L´evy processes in groups and symmetric spaces. J. Theoret. Probab. To appear.
  • [3] Applebaum,D. and Kunita,H. (1993). L´evy flows on manifolds and L´evy processes on Lie groups. J. Math. Kyoto Univ. 33 1103-23.
  • [4] Bertoin,J. (1996). L´evy Processes. Cambridge Univ. Press.
  • [5] Cohen,S. (1996). G´eometrie diff´erentielle avec sauts I. Stochastics Stochastics Rep. 56 179- 203.
  • [6] Elworthy,K. D. (1982). Stochastic Differential Equations on Manifolds. Cambridge Univ. Press.
  • [7] Elworthy,K.D. (1988). Geometric aspects of diffusions on manifolds. Ecole d'Et´e de Probabilit´es de Saint-Flour XVII. Lecture Notes in Math. 1362 276-425. Springer, Berlin.
  • [8] Emery,M. (1989). Stochastic Calculus in Manifolds. Springer, Berlin.
  • [9] Estrade,A. and Pontier,M. (1992). Rel evement horizontal d'une semi-martingale cadlag. S´eminaire de Probabilit´es XXVI. Lecture Notes in Math. 1526 127-145. Springer, Berlin.
  • [10] Fujiwara,T. (1991). Stochastic differential equations of jump type on manifolds and L´evy flows. J. Math. Kyoto Univ. 31 99-119.
  • [11] Gangolli,R. (1964). Isotropic infinitely divisible measures on symmetric spaces. Acta Math. 111 213-246.
  • [12] Gangolli,R. (1965). Sample functions of certain differential processes on symmetric spaces. Pacific J. Math. 15 477-496.
  • [13] Helgason,S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, NewYork.
  • [14] Hunt,G. A. (1956). Semigroups of measures on Lie groups. Trans. Amer. Math. Soc. 81 264-293.
  • [15] Ikeda,N. and Watanabe,S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • [16] It o,K. (1942). On stochastic processes I (infinitely divisible laws of probability). Japan. J. Math. 18 261-301.
  • [17] Jacod,J. and Shiryaev,A. N. (1987). Limit Theorems for Stochastic Processes. Springer, NewYork.
  • [18] Kobayashi,S. and Nomizu,K. (1963). Foundations of Differential Geometry 1. Wiley, New York.
  • [19] Kunita,H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press.
  • [20] Kunita,H. (1996). Stochastic differential equations with jumps and stochastic flows of diffeomorphisms. In It o's Stochastic Calculus and Probability Theory (N. Ikeda, S. Wantanabe, M. Fukushima and H. Konita, eds.) 197-212. Springer, Berlin.
  • [21] Malliavin,P. (1997). Stochastic Analysis. Springer, Berlin.
  • [22] Revuz,D. and Yor,M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.