Take the usual Brownian motion on $R^d$ and project it to $T^d$, for almost every individual trajectory, will it be equidistributed on the torus? Does this depend on $d$?



Is a typical path of a Brownian motion on a torus equidistributed?
Yes - it is equidistributed (which is the case for the Brownian motion on any compact Riemannian manifold).


