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A two-dimensional random crystalline algorithm for Gauss curvature flow

Part of: Geometric probability and stochastic geometry Markov processes

Published online by Cambridge University Press:  01 July 2016

H. Ishii  and
T. Mikami
H. Ishii*
Affiliation:
Waseda University
T. Mikami*
Affiliation:
Hokkaido University
*
∗ Postal address: Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishiwaseda, Shinjyuku-ku, Tokyo 169-8050, Japan.
∗∗ Postal address: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan. Email address: mikami@math.sci.hokudai.ac.jp
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Abstract

We propose and study a random crystalline algorithm (a discrete approximation) of the Gauss curvature flow of smooth simple closed convex curves in ℝ2 as a stepping stone to the full understanding of such phenomena as the wearing process of stones on a beach.

Keywords

Random crystalline algorithmGauss curvature flowclosed curve

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J75: Jump processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 491 - 504
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1] Chow, B. (1985). Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J. Differential Geometry 22, 117–138.Google Scholar
[2] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
[3] Firey, W. J. (1974). Shapes of worn stones. Mathematika 21, 1–11.Google Scholar
[4] Girão, P. M. (1995). Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature. SIAM J. Numer. Anal. 32, 886–899.Google Scholar
[5] Girão, P. M. and Kohn, R. V. (1994). Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature. Numer. Math. 67, 41–70.Google Scholar
[6] Huisken, G. (1984). Flow by mean curvature of convex surfaces into spheres. J. Differential Geometry 20, 237–266.Google Scholar
[7] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland/Kodansha, Tokyo.Google Scholar
[8] Ishii, H. (1999). An approximation scheme for Gauss curvature flow and its convergence. Preprint.Google Scholar
[9] Ishii, K. and Soner, H. M. (1998). Regularity and convergence of crystalline motion. SIAM J. Math. Anal. 30, 19–37.Google Scholar
[10] Meyer, P. A. (1976). Un cours sur les intégrales stochastiques. In Séminaire de Probabilités X (Lecture Notes Math. 511), ed. Meyer, P. A., Springer, Berlin, pp. 245–400.Google Scholar
[11] Tso, K. (1985). Deforming a hypersurface by its Gauss–Kronecker curvature. Commun. Pure Appl. Math. 38, 867–882.Google Scholar