  We show that submanifolds with infinite mean exit time can not be
isometrically and minimally immersed into cylinders, horocylinders, cones, and
wedges of some product spaces. Our approach is not based on the weak maximum
principle at infinity, and thus it permits us to generalize previous results
concerning non-immersibility of stochastically complete submanifolds. We also
produce estimates for the complete tower of moments for submanifolds with small
mean curvature immersed into cylinders.
