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Stochastic approximations to curve-shortening flows via particle systems. (English) Zbl 1032.60092

Summary: 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. 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
35K55 Nonlinear parabolic equations
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