Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

• 成果类型: Article
• 署名作者: Hofmanova, Martina; Roeger, Matthias; von Renesse, Max
• 署名单位: Max Planck Society; Technical University of Berlin; Dortmund University of Technology; Leipzig University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0713-5
• 发表日期: 2017
• 页码: 373-408
• 关键词: viscosity solutions; level sets; motion; uniqueness; EQUATIONS; EXISTENCE; pde

摘要

We study a stochastically perturbed mean curvature flow for graphs in over the two-dimensional unit-cube subject to periodic boundary conditions. The stochastic perturbation is a one dimensional white noise acting uniformly in all points of the surface in normal direction. We establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods.