## Curvature conditions on Riemannian manifolds with Brownian harmonicity properties

HTML articles powered by AMS MathViewer

- by H. R. Hughes PDF
- Trans. Amer. Math. Soc.
**347**(1995), 339-361 Request permission

## 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.## References

- Arthur L. Besse,
*Manifolds all of whose geodesics are closed*, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR**496885**, DOI 10.1007/978-3-642-61876-5
R. Courant and D. Hilbert, - Ewa Damek and Fulvio Ricci,
*Harmonic analysis on solvable extensions of $H$-type groups*, J. Geom. Anal.**2**(1992), no. 3, 213–248. MR**1164603**, DOI 10.1007/BF02921294 - Alfred Gray and Mark A. Pinsky,
*The mean exit time from a small geodesic ball in a Riemannian manifold*, Bull. Sci. Math. (2)**107**(1983), no. 4, 345–370 (English, with French summary). MR**732357** - H. R. Hughes,
*Brownian exit distributions from normal balls in $S^3\times H^3$*, Ann. Probab.**20**(1992), no. 2, 655–659. MR**1159565**, DOI 10.1214/aop/1176989797 - Nobuyuki Ikeda and Shinzo Watanabe,
*Stochastic differential equations and diffusion processes*, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR**1011252**
K. Itô and H. P. McKean, Jr., - Oldřich Kowalski,
*The second mean-value operator on Riemannian manifolds*, Proceedings of the Conference on Differential Geometry and its Applications (Nové Město na Moravě, 1980) Univ. Karlova, Prague, 1982, pp. 33–45. MR**663211** - Masanori K\B{o}zaki and Yukio Ogura,
*On geometric and stochastic mean values for small geodesic spheres in Riemannian manifolds*, Tsukuba J. Math.**11**(1987), no. 1, 131–145. MR**899727**, DOI 10.21099/tkbjm/1496160508 - Masanori K\B{o}zaki and Yukio Ogura,
*On the independence of exit time and exit position from small geodesic balls for Brownian motions on Riemannian manifolds*, Math. Z.**197**(1988), no. 4, 561–581. MR**932686**, DOI 10.1007/BF01159812 - Ming Liao,
*Hitting distributions of small geodesic spheres*, Ann. Probab.**16**(1988), no. 3, 1039–1050. MR**942754** - Ming Liao,
*An independence property of Riemannian Brownian motions*, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 197–201. MR**954638**, DOI 10.1090/conm/073/954638 - Mark Pinsky,
*Moyenne stochastique sur une variété riemannienne*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 23, 991–994 (French, with English summary). MR**630934** - Mark A. Pinsky,
*On non-Euclidean harmonic measure*, Ann. Inst. H. Poincaré Probab. Statist.**21**(1985), no. 1, 39–46 (English, with French summary). MR**791268** - H. S. Ruse, A. G. Walker, and T. J. Willmore,
*Harmonic spaces*, Consiglio Nazionale delle Ricerche Monografie Matematiche, vol. 8, Edizioni Cremonese, Rome, 1961. MR**0142062** - N. Ja. Vilenkin and A. U. Klimyk,
*Representation of Lie groups and special functions. Vol. 2*, Mathematics and its Applications (Soviet Series), vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1993. Class I representations, special functions, and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR**1220225**, DOI 10.1007/978-94-017-2883-6

*Methods of mathematical physics*, vol. 2, Interscience, New York, 1962.

*Diffusion processes and their sample paths*, Grundlehren Math. Wiss., 125, Springer, Berlin, 1965.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**347**(1995), 339-361 - MSC: Primary 58G32; Secondary 53C21, 60J65
- DOI: https://doi.org/10.1090/S0002-9947-1995-1276934-8
- MathSciNet review: 1276934