  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.
