Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

In this paper, we deal with stochastically complete submanifolds M n immersed with nonzero parallel mean curvature vector field in a Riemannian space form Q(c)(n+ p) of constant sectional curvature c is an element of {-1, 0, 1}. In this setting, we use the weak Omori-Yau maximum principle jointly with a suitable Simons type formula in order to show that either such a submanifold M-n must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only if the submanifold is isometric to an open piece of a hyperbolic cylinder H-1 ( -root 1 + r(2) ) x Sn-1 (r), when c = -1, a circular cylinder R x Sn-1 (r), when c = 0, and a Clifford torus S-1 (root 1 - r(2) ) x Sn-1 (r), when c = 1.