A stochastically perturbed mean curvature flow by colored noise

2018/11/10 by Satoshi Yokoyama, Yokoyama, Satoshi
Economics, Econometrics and Finance · Engineering · Mathematics · #35K93 #60H15 #74A50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1811.04265

openalex publication_date 2018/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the motion of the hypersurface (γ_t)_{t≥ 0} evolving according to the mean curvature perturbed by w^Q, the formal time derivative of the Q-Wiener process w^Q, in a two dimensional bounded domain. Namely, we consider the equation describing the evolution of γ_t as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity V is determined by V=κ + G ∘ w^Q, where κ is the mean curvature and G is a function determined from γ_t. Already known results in which the noise depends on only time variable is not applicable to our equation. To construct a local solution of the equation describing γ_t, we will derive a certain second order quasilinear SPDE with respect to the signed distance function determined from γ₀. Then we construct the local solution making use of probabilistic tools and the classical Banach fixed-point theorem on suitable Sobolev spaces.

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