We study the regularity of the sample paths of certain
Brownian motions on the
infinite dimensional torus ${\mathbb T}^\infty$ and other compact
connected groups in terms of the associated intrinsic distance.
For each $\lambda\in (0,1)$,
we give examples where the intrinsic distance $d$ is continuous and
defines the topology of ${\mathbb T}^\infty$ and where the
sample paths satisfy
\[
0<\liminf_{t\ra 0}
\frac{d(X_0,X_t)}{t^{(1-\lambda)/2}}\le
\limsup_{t\ra 0}
\frac{d(X_0,X_t)}{t^{(1-\lambda)/2}}<\infty
\]
and
\[
0<\lim_{\varepsilon\to 0}
\sup_{0
@article{1055425787,
author = {Bendikov, Alexander and Saloff-Coste, Laurent},
title = {On the sample paths of Brownian motions on compact infinite dimensional groups},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 1464-1493},
language = {en},
url = {http://dml.mathdoc.fr/item/1055425787}
}
Bendikov, Alexander; Saloff-Coste, Laurent. On the sample paths of Brownian motions on compact infinite dimensional groups. Ann. Probab., Tome 31 (2003) no. 1, pp. 1464-1493. http://gdmltest.u-ga.fr/item/1055425787/