The Annals of Probability

Range of fluctuation of Brownian motion on a complete Riemannian manifold

Alexander Grigor'yan and Mark Kelbert

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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 $\varepsilon > 0$).

Article information

Ann. Probab., Volume 26, Number 1 (1998), 78-111.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 58G32 58G11
Secondary: 60G17: Sample path properties 60F15: Strong theorems

Brownian motion heat kernel Riemannian manifold escape rate the law of the iterated logarithm


Grigor'yan, Alexander; Kelbert, Mark. Range of fluctuation of Brownian motion on a complete Riemannian manifold. Ann. Probab. 26 (1998), no. 1, 78--111. doi:10.1214/aop/1022855412.

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