Almost sure asymptotic behaviour of the number of windings of the Brownian motion of a compact Riemannian manifold of dimension 2 or 3 (Q1897522)

Let \(M\) be a connected and compact smooth Riemannian manifold of dimension 2 and let \(X\) be a Brownian motion on \(M\). For a closed differential form \(\omega\) on \(M\) with a finite set of singularities we define \[ N_t = {1\over t} \int^t_0 \omega (X_s), \] where the integral is the Stratonovich integral along the paths of \(X\). Then for an increasing deterministic function \(f\) the following holds. We have \(\text{limsup} |N_t|/f(t) = 0\) or \(\infty\) if and only if \(\int^\infty \text{dt}/f(t) < \infty\) or \(\infty\). A generalization to the dimension 3 is also given.