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January 1998 Range of fluctuation of Brownian motion on a complete Riemannian manifold
Alexander Grigor'yan, Mark Kelbert
Ann. Probab. 26(1): 78-111 (January 1998). DOI: 10.1214/aop/1022855412

Abstract

We investigate the escape rate of the Brownian motion $W_x (t)$ on a complete noncompact Riemannian manifold. Assuming that the manifold has at most polynomial volume growth and that its Ricci curvature is bounded below, we prove that $$\dist (W_x (t), x) \leq \sqrt{Ct \log t}$$ for all large $t$ with probability 1. On the other hand, if the Ricci curvature is nonnegative and the volume growth is at least polynomial of the order $n > 2$ then $$\dist (W_x (t), x) \geq \frac{\sqrt{Ct}}{\log^{1/(n-2)} t \log \log^{(2+\varepsilon)/(n-2)} t} again for all large $t$ with probability 1 (where $\varepsilon > 0$).

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Alexander Grigor'yan. Mark Kelbert. "Range of fluctuation of Brownian motion on a complete Riemannian manifold." Ann. Probab. 26 (1) 78 - 111, January 1998. https://doi.org/10.1214/aop/1022855412

Information

Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0934.58023
MathSciNet: MR1617042
Digital Object Identifier: 10.1214/aop/1022855412

Subjects:
Primary: 58G11, 58G32
Secondary: 60F15, 60G17

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.26 • No. 1 • January 1998
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