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A Stochastically Perturbed Mean Curvature Flow by Colored Noise

  • Satoshi YokoyamaEmail author
Article

Abstract

We study the motion of the hypersurface \((\gamma _t)_{t\ge 0}\) evolving according to the mean curvature perturbed by \(\dot{w}^Q\), 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 \(\gamma _t\) as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity V is determined by \(V=\kappa \,+\,G \circ \dot{w}^Q\), where \(\kappa \) is the mean curvature and G is a function determined from \(\gamma _t\). Already known results in which the noise depends on only the time variable are not applicable to our equation. To construct a local solution of the equation describing \(\gamma _t\), we derive a certain second-order quasilinear SPDE with respect to the signed distance function determined from \(\gamma _0\). Then we construct the local solution making use of probabilistic tools and the classical Banach fixed point theorem on suitable Sobolev spaces.

Keywords

Mean curvature flow Stochastic perturbation Colored noise 

Mathematics Subject Classification

60H15 35K93 74A50 

Notes

Acknowledgements

This research is supported by JSPS KIKIN Grant 18K13430. The author greatly thanks referees for the kindest comments. Also he would like to thank Professor Martina Hofmanovà, Professor Tadahisa Funaki and Pierre Simonot for helpful discussions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of Mathematics, School of Fundamental Science and EngineeringWaseda UniversityTokyoJapan

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