On stochastically complete submanifolds

Abstract

Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional curvature estimates for properly immersed cilindrically bounded submanifolds.

This paper has been withdrawn by the authors. The theorem that we seemed to prove is false. There are stochastically incomplete properly immersed in ${\mathbb{R}}^n$