On the sample paths of Brownian motions on compact infinite dimensional groups (Q1431499)

The authors study the regularity of the paths of certain Brownian motions on the infinite-dimensional torus and other compact connected groups in terms of the intrinsic distance \(d\) on the group. In particular, for each \(\lambda\in ]0,1[\), examples of such processes \((X_t)_{t\geq 0}\) are constructed such that, for \(t\to 0\), \(d(X_t, X_0)\) roughly behaves like \(t^{(1-\lambda)/2}\) almost surely. Moreover, an associated result on the modulus of continuity is derived. These results are quite different to the case of finite-dimensional Lie groups where, independent of the dimension like for the classical Brownian motion, a behavior of order \(\sqrt{4t\ln(1/t)}\) holds.