Products of constant curvature spaces with a Brownian independence property

Hughes, H.

doi:10.1090/S0002-9939-98-04447-5

Products of constant curvature spaces with a Brownian independence property
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by H. R. Hughes

Proc. Amer. Math. Soc. 126 (1998), 3417-3425

DOI: https://doi.org/10.1090/S0002-9939-98-04447-5

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Abstract:

The time and place Brownian motion on the product of constant curvature spaces first exits a normal ball of radius $\epsilon$ centered at the starting point of the Brownian motion are considered. The asymptotic expansions, as $\epsilon$ decreases to zero, for joint moments of the first exit time and place random variables are computed with error $O(\epsilon ^{10})$. It is shown that the first exit time and place are independent random variables only if each factor space is locally flat or of dimension three.

References

R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360
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
H. R. Hughes, Curvature conditions on Riemannian manifolds with Brownian harmonicity properties, Trans. Amer. Math. Soc. 347 (1995), no. 1, 339–361. MR 1276934, DOI 10.1090/S0002-9947-1995-1276934-8
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
Masanori K\B{o}zaki and Yukio Ogura, Riemannian manifolds with stochastic independence conditions are rich enough, Probability theory and mathematical statistics (Kyoto, 1986) Lecture Notes in Math., vol. 1299, Springer, Berlin, 1988, pp. 206–213. MR 935991, DOI 10.1007/BFb0078475
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
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