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

• 成果类型: Article
• 署名作者: Dabrock, Nils; Hofmanova, Martina; Roger, Matthias
• 署名单位: Dortmund University of Technology; University of Bielefeld
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01012-6
• 发表日期: 2021
• 页码: 407-449
• 关键词: weak solutions; level sets; front propagation; wave-equations; motion; uniqueness; Discretization; EVOLUTION; pde

摘要

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-w,x,t(infinity) estimate for the gradient and an L-w, 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.