## The Annals of Probability

### On the sample paths of Brownian motions on compact infinite dimensional groups

#### Abstract

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 $0<\lim_{\varepsilon\to 0} \sup_{0<t<s<1 \atop t-s\le \varepsilon}\frac{d(X_s,X_t)}{(t-s )^{(1-\lambda)/2}} <\infty.$

#### Article information

Source
Ann. Probab., Volume 31, Number 3 (2003), 1464-1493.

Dates
First available in Project Euclid: 12 June 2003

https://projecteuclid.org/euclid.aop/1055425787

Digital Object Identifier
doi:10.1214/aop/1055425787

Mathematical Reviews number (MathSciNet)
MR1989440

Zentralblatt MATH identifier
1043.60064

#### Citation

Bendikov, Alexander; Saloff-Coste, Laurent. On the sample paths of Brownian motions on compact infinite dimensional groups. Ann. Probab. 31 (2003), no. 3, 1464--1493. doi:10.1214/aop/1055425787. https://projecteuclid.org/euclid.aop/1055425787

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• ITHACA, NEW YORK 14853-4201 E-MAIL: lsc@math.cornell.edu bendikov@math.cornell.edu