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