  We are concerned with a stochastic mean curvature flow of graphs over a
periodic domain of any space dimension. We establish existence of martingale
solutions which are strong in the PDE sense and study their large-time
behavior. Our analysis is based on a viscous approximation and new global
bounds, namely, an $L^{\infty}_{\omega,x,t}$ estimate for the gradient and an
$L^{2}_{\omega,x,t}$ bound for the Hessian. The proof makes essential use of
the delicate interplay between the deterministic mean curvature part and the
stochastic perturbation, which permits to show that certain gradient-dependent
energies are supermartingales. Our energy bounds in particular imply that
solutions become asymptotically spatially homogeneous and approach a Brownian
motion perturbed by a random constant.
