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Transactions of the American Mathematical Society

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



Curvature conditions on Riemannian manifolds with Brownian harmonicity properties

Author: H. R. Hughes
Journal: Trans. Amer. Math. Soc. 347 (1995), 339-361
MSC: Primary 58G32; Secondary 53C21, 60J65
MathSciNet review: 1276934
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Abstract: The time and place that Brownian motion on a Riemannian manifold first exits a normal ball of radius $ \varepsilon $ is considered and a general procedure is given for computing asymptotic expansions, as $ \varepsilon $ decreases to zero, for joint moments of the first exit time and place random variables. It is proven that asymptotic versions of exit time and place distribution properties that hold on harmonic spaces are equivalent to certain curvature conditions for harmonic spaces. In particular, it is proven that an asymptotic mean value condition involving first exit place is equivalent to certain levels of curvature conditions for harmonic spaces depending on the order of the asymptotics. Also, it is proven that an asymptotic uncorrelated condition for first exit time and place is equivalent to weaker curvature conditions at corresponding orders of asymptotics.

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  • [B] A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., vol. 93, Springer, Berlin, 1978. MR 496885 (80c:53044)
  • [CH] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 2, Interscience, New York, 1962.
  • [DR] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of $ H$-type groups, J. Geom. Anal. 2 (1992), 213-248. MR 1164603 (93d:43006)
  • [GP] A. Gray and M. Pinsky, The mean exit time from a small geodesic ball in a Riemannian manifold, Bull. Sci. Math. (2) 107 (1983), 345-370. MR 732357 (85b:58128)
  • [Hu] H. R. Hughes, Brownian exit distributions from normal balls in $ {S^3} \times {H^3}$, Ann. Probab. 20 (1992), 655-659. MR 1159565 (93e:58196)
  • [IW] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland, Amsterdam, 1989. MR 1011252 (90m:60069)
  • [IM] K. Itô and H. P. McKean, Jr., Diffusion processes and their sample paths, Grundlehren Math. Wiss., 125, Springer, Berlin, 1965.
  • [K] O. Kowalski, The second mean-value operator on Riemannian manifolds, Proceedings of the CSSR-GDR-Polish conference on differential geometry and its applications, Nové Město 1980, Universita Karlova Praha, 1982, pp. 33-45. MR 663211 (83j:53035)
  • [KO1] M. Kozaki and Y. Ogura, On geometric and stochastic mean values for small geodesic spheres in Riemannian manifolds, Tsukuba J. Math. 11 (1987), 131-145. MR 899727 (89b:53085)
  • [KO2] -, On the independence of exit time and exit position from small geodesic balls for Brownian motions on Riemannian manifolds, Math. Z. 197 (1988), 561-581. MR 932686 (89g:58218)
  • [L1] M. Liao, Hitting distributions of small geodesic spheres, Ann. Probab. 16 (1988), 1039-1050. MR 942754 (89g:58219)
  • [L2] -, An independence property of Riemannian Brownian motions, Geometry of Random Motion, Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 197-201. MR 954638 (89j:58143)
  • [P1] M. Pinsky, Moyenne stochastique sur une variété riemannienne, C. R. Acad. Sci. Paris 292 (1981), 991-994. MR 630934 (84f:58117)
  • [P2] -, On non-Euclidean harmonic measure, Ann. Inst. H. Poincaré 21 (1985), 39-46. MR 791268 (86m:58152)
  • [RWW] H. Ruse, H. G. Walker, and T. J. Willmore, Harmonic spaces, Monogr. Mat., 8, Edizioni Cremonese, Roma, 1961. MR 0142062 (25:5456)
  • [VK] N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions, Volume 2: class I representations, special functions, and integral transforms, Mathematics and its Applications (Soviet Series), vol. 74, Kluwer, Dordrecht, 1993. MR 1220225 (94m:22001)

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Keywords: Brownian motion, harmonic space, curvature conditions, exit time, exit place
Article copyright: © Copyright 1995 American Mathematical Society

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