Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

Abstract

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $$L^{\infty }_{\omega ,x,t}$$ L ω , x , t ∞ estimate for the gradient and an $$L^{2}_{\omega ,x,t}$$ L ω , x , t 2 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.

The Neumann problem for fully nonlinear SPDE

“…In fact, our proof crucially relies on their result, which we combine with the stability properties of the level set PDE to obtain our convergence (in particular, both the spatial modulus and monotonicity results are needed). Similar results have also been obtained in the stochastic PDE literature in the simpler case of periodic graphs and by different methods in [ESvR12,DHR21].…”