TY - GEN

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

AU - Denis, Clément

AU - Funaki, Tadahisa

AU - Yokoyama, Satoshi

N1 - Publisher Copyright:
© Springer International Publishing AG, part of Springer Nature 2018.

PY - 2018

Y1 - 2018

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

AN - SCOPUS:85049969274

SN - 9783319749280

T3 - Springer Proceedings in Mathematics and Statistics

SP - 177

EP - 200

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

A2 - Trutnau, Gerald

A2 - Eberle, Andreas

A2 - Hoh, Walter

A2 - Kassmann, Moritz

A2 - Grothaus, Martin

A2 - Stannat, Wilhelm

PB - Springer New York LLC

T2 - International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016

Y2 - 10 October 2016 through 14 October 2016

ER -