Transience and capacity of minimal submanifolds (Q1413659)

It is well known that the Brownian motion of the particle depends upon the topology and dimension of the manifold on which such motion takes place. The simplest examples of such dependence are 1, 2 and 3 dimensional Euclidean spaces for which the Brownian motion is recurrent for dimensions 1 and 2 and transient for 3 dimensions. The present article deals with the extension of these results for Riemannian manifolds and submanifolds. The authors prove transience for complete minimally immersed \(m\)-dimensional submanifolds \(P(m)\) of Hadamard-Cartan manifolds \(N(n)\) (of dimension \(n\)) whose sectional curvature is bounded from above by \(b\) less or equal to zero. It is proven that Brownian motion on \(P(m)\) is transient if either \(b<0\) and \(m\) greater or equal to 2 or \(b=0\) and \(m\) greater and equal to 3.