Liouville property and the linear drift of Brownian motion (Q2370797)

Let \((M,g)\) be a complete connected Riemannian manifold. Assume that \(M\) is a regular covering of a Riemannian manifold of finite volume. Then the authors obtained a neat characterization for \(M\) being Liouville by means of the Brownian motion \(B_t\). More precisely, if \(M\) admits bounded sectional curvature, then \(M\) is Liouville if and only if \(\lim_{t\rightarrow +\infty}\tfrac1t\,d(x_0, B_t)=0\) a.s. As explained by the authors, some conditions concerning the symmetry are essential to obtain such neat criterion.