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Curvature Motion Perturbed by a Direction-Dependent Colored Noise

  • Clément Denis
  • Tadahisa FunakiEmail author
  • Satoshi Yokoyama
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

The aim of this paper is twofold. First we give a brief overview of several results on the deterministic and stochastic motions by mean curvature and their derivation under the so-called sharp interface limit. Then, we study the motions by mean curvature perturbed by a direction-dependent Gaussian colored noise described by \(V=\kappa + \dot{W}(t,\mathbf n )\). This part is a generalization of (Funaki, Acta Math Sin (Engl Ser), 15:407–438, 1999) [10] where the noise is independent from space. We derive a uniform moment estimate on solutions of approximating equations and prove a Wong–Zakai type convergence theorem (in law) for the SPDEs for the curvature of a convex curve in two-dimensional space before the time the curve exhibits a singularity.

Keywords

Stochastic partial differential equation Motion by mean curvature Wong–Zakai theorem Colored noise 

MSC 2010

60H15 35K93 

Notes

Acknowledgements

We thank the referee who suggested the proof of Theorem 1, in particular, the SPDE (26) and the reference [5]. We acknowledge the support from the training course at ENS Cachan, under which C. Denis could visit Tokyo and stayed for four months in 2016.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Clément Denis
    • 1
  • Tadahisa Funaki
    • 2
    • 3
    Email author
  • Satoshi Yokoyama
    • 2
    • 3
  1. 1.Département de mathématiquesEcole Normale Supérieure (ENS) de Paris -SaclayCachanFrance
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan
  3. 3.Department of Mathematics, School of Fundamental Science and EngineeringWaseda UniversityTokyoJapan

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