Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Isotropic transport process on a Riemannian manifold

Author: Mark A. Pinsky
Journal: Trans. Amer. Math. Soc. 218 (1976), 353-360
MSC: Primary 60J65; Secondary 58G99
MathSciNet review: 0402957
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class $ {C_0}$. The special case of a manifold of negative curvature is considered as an illustration.

References [Enhancements On Off] (What's this?)

  • [1] Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 0356254
  • [2] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
  • [3] A. Debiard, B. Gaveau and E. Mazet, Temps d'arrêt des diffusions riemanniennes, C. R. Acad. Sci. Paris Sér. A.-B 278 (1974), A723-A725. MR 49 #6381a.
  • [4] Ramesh Gangolli, On the construction of certain diffusions on a differentiable manifold, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 406–419 (1964). MR 0165590,
  • [5] R. Goldberg, Curvature and homology, Academic Press, New York, 1968.
  • [6] Noel J. Hicks, Notes on differential geometry, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0179691
  • [7] Kiyosi Itô, Stochastic differentials of continuous local quasi-martingales, Stability of stochastic dynamical systems (Proc. Internat. Sympos., Univ. Warwick, Coventry, 1972) Springer, Berlin, 1972, pp. 1–7. Lecture Notes in Math., Vol. 294. MR 0405583
  • [8] Thomas G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis 12 (1973), 55–67. MR 0365224
  • [9] Thomas G. Kurtz, Semigroups of conditioned shifts and approximation of Markov processes, Ann. Probability 3 (1975), no. 4, 618–642. MR 0383544
  • [10] Daniel W. Stroock, On the growth of stochastic integrals, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 340–344. MR 0287622,
  • [11] Shinzo Watanabe and Toitsu Watanabe, Convergence of isotropic scattering transport process to Brownian motion, Nagoya Math. J. 40 (1970), 161–171. MR 0279885
  • [12] Erik Jørgensen, The central limit problem for geodesic random walks, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 1–64. MR 0400422,
  • [13] P. Malliavin, Diffusions et géométrie différentielle globale, Lecture Notes, August 1975, Institut Henri Poincaré, 11 rue Pierre et Marie Curie, Paris 5.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J65, 58G99

Retrieve articles in all journals with MSC: 60J65, 58G99

Additional Information

Keywords: Isotropic transport process, geodesic flow, Brownian motion, contraction semigroup
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society