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.
CITATION STYLE
Denis, C., Funaki, T., & Yokoyama, S. (2018). Curvature motion perturbed by a direction-dependent colored noise. In Springer Proceedings in Mathematics and Statistics (Vol. 229, pp. 177–200). Springer New York LLC. https://doi.org/10.1007/978-3-319-74929-7_9
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