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 $\varepsilon > 0$).
Publié le : 1998-01-14
Classification:
Brownian motion,
heat kernel,
Riemannian manifold,
escape rate,
the law of the iterated logarithm,
58G32,
58G11,
60G17,
60F15
@article{1022855412,
author = {Grigor'yan, Alexander and Kelbert, Mark},
title = {Range of fluctuation of Brownian motion on a complete Riemannian
manifold},
journal = {Ann. Probab.},
volume = {26},
number = {1},
year = {1998},
pages = { 78-111},
language = {en},
url = {http://dml.mathdoc.fr/item/1022855412}
}
Grigor'yan, Alexander; Kelbert, Mark. Range of fluctuation of Brownian motion on a complete Riemannian
manifold. Ann. Probab., Tome 26 (1998) no. 1, pp. 78-111. http://gdmltest.u-ga.fr/item/1022855412/