Abstract
Curvature driven flows have been extensively considered from a deterministic point of view. Besides their mathematical interest, they have been shown to be useful for a number of applications including crystal growth, flame propagation, and computer vision. In this paper, we describe a random particle system, evolving on the discretized unit circle, whose profile converges toward the Gauss-Minkowsky transformation of solutions of curve shortening flows initiated by convex curves. Our approach may be considered as a type of stochastic crystalline algorithm. Our proofs are based on certain techniques from the theory of hydrodynamical limits.
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References
S. Angenent, G. Sapiro, and A. Tannenbaum, On the affine heat equation for non-convex curves, Journal of the American Mathematical Society 11 (1998), pp. 601–634.
B. Andrews, Evolving convex curves, Calc. Var. 7 (1998), pp. 315–371.
B. Andrews, Non-convergence and instability in the limiting behaviour of curves evolving by curvature, to appear in Communications in Analysis and Geometry.
G. Ben Arous, A. Tannenbaum, and O. Zeitouni, Stochastic approximations to curve shortening flows via particle systems, Technical Report, February 2002. Preprint may be found at http://www.ee.technion.ac.il/~zeitouni/ps/hydro6.ps.
H. Buseman, Convex surfaces, Interscience Publ. (1958).
K.S. Chow and X.P. Zhu, The curve shortening problem, Chapman-Hall/CRC (2001).
P. Del Moral and L. Miclo, Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering, Séminaire de Probabilités XXXIV, Lecture Notes in Math. 1729, Springer (2000), pp. 1–145.
M. Gage and R.S. Hamilton, The heat equation shrinking convex planar curves, J. Diff. Geom. 23 (1986), pp. 69–96.
M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geometry 26 (1987), pp. 285–314.
S. Haker, G. Sapiro, and A. Tannenbaum, Knowledge-based segmentation of SAR data with learned priors, IEEE Trans. Image Processing 9 (2000), pp. 298–302.
C. Kipnis and C. Landim, Scaling limits of interacting particle systems, Springer (1999).
G.M. Lieberman, Second order parabolic differential equations, World Scientific (1996).
M. Mourragui, Comportement hydrodynamique et entropie relative des processus de sauts, de naissances et de morts, Ann. Inst. H. Poincaré 32 (1996), pp. 361–385.
P. Olver, G. Sapiro, and A. Tannenbaum, Differential invariant signatures and flows in computer vision: a symmetry group approach, in Geometry Driven Diffusion in Computer Vision, edited by Bart Romeny, Kluwer, Holland, 1994.
M.H. Protter and H.F. Weinberger, Maximum principles in differential equations, Springer (1984).
G. Sapiro and A. Tannenbaum, On affine planar curve evolution, J. Functl. Anal 119 (1994), pp. 79–120.
H. Taniyama and H. Matano, Formation of singularities in general curve shortening equations, preprint.
T.K. Ushijima and S. Yazaki, Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature V = K α, SIAM. J. Numer. Anal 37 (2000), pp. 500–522.
A. Vershik and O. Zeitouni, Large deviations in the geometry of convex lattice polygons, Isr. J. Math. 109 (1999), pp. 13–27.
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Arous, G.B., Tannenbaum, A., Zeitouni, O. (2003). Crystalline Stochastic Systems and Curvature Driven Flows. In: Rosenthal, J., Gilliam, D.S. (eds) Mathematical Systems Theory in Biology, Communications, Computation, and Finance. The IMA Volumes in Mathematics and its Applications, vol 134. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21696-6_2
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DOI: https://doi.org/10.1007/978-0-387-21696-6_2
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