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Curvature Motion Perturbed by a Direction-Dependent Colored Noise

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Stochastic Partial Differential Equations and Related Fields (SPDERF 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

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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.

Author information

Authors and Affiliations

  1. Département de mathématiques, Ecole Normale Supérieure (ENS) de Paris -Saclay, 61 avenue du Président Wilson, Cachan, France

    Clément Denis

  2. Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153 -8914, Japan

    Tadahisa Funaki & Satoshi Yokoyama

  3. Department of Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan

    Tadahisa Funaki & Satoshi Yokoyama

Authors
  1. Clément Denis
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  2. Tadahisa Funaki
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  3. Satoshi Yokoyama
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Corresponding author

Correspondence to Tadahisa Funaki .

Editor information

Editors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Andreas Eberle

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Martin Grothaus

  3. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Walter Hoh

  4. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Moritz Kassmann

  5. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Wilhelm Stannat

  6. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea (Republic of)

    Gerald Trutnau

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Denis, C., Funaki, T., Yokoyama, S. (2018). Curvature Motion Perturbed by a Direction-Dependent Colored Noise. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_9

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