### 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=κ + W(t, 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.

Original language | English |
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Title of host publication | Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016 |

Publisher | Springer New York LLC |

Pages | 177-200 |

Number of pages | 24 |

Volume | 229 |

ISBN (Print) | 9783319749280 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

Event | International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 - Bielefeld, Germany Duration: 2016 Oct 10 → 2016 Oct 14 |

### Other

Other | International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 |
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Country | Germany |

City | Bielefeld |

Period | 16/10/10 → 16/10/14 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016*(Vol. 229, pp. 177-200). Springer New York LLC. https://doi.org/10.1007/978-3-319-74929-7_9

**Curvature motion perturbed by a direction-dependent colored noise.** / Denis, Clément; Funaki, Tadahisa; Yokoyama, Satoshi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016.*vol. 229, Springer New York LLC, pp. 177-200, International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016, Bielefeld, Germany, 16/10/10. https://doi.org/10.1007/978-3-319-74929-7_9

}

TY - GEN

T1 - Curvature motion perturbed by a direction-dependent colored noise

AU - Denis, Clément

AU - Funaki, Tadahisa

AU - Yokoyama, Satoshi

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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=κ + W(t, 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.

AB - 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=κ + W(t, 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.

KW - Colored noise

KW - Motion by mean curvature

KW - Stochastic partial differential equation

KW - Wong–Zakai theorem

UR - http://www.scopus.com/inward/record.url?scp=85049969274&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049969274&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-74929-7_9

DO - 10.1007/978-3-319-74929-7_9

M3 - Conference contribution

SN - 9783319749280

VL - 229

SP - 177

EP - 200

BT - Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016

PB - Springer New York LLC

ER -