Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T19:49:14.283Z Has data issue: false hasContentIssue false
  • English
  • Français

A two-dimensional random crystalline algorithm for Gauss curvature flow

Part of: Markov processes Geometric probability and stochastic geometry

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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