Arnaudon, Marc, Coulibaly-Pasquier, Abdoulaye Koléhè and Miclo, Laurent (2024) The stochastic renormalized mean curvature flow for planar convex sets. Electronic Journal of Probability, vol. 29 (n° 183). pp. 1-43. (In Press)

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Abstract

We investigate renormalized curvature flow (RCF) and stochastic renormalized curvature flow (SRCF) for convex sets in the plane. RCF is the gradient descent flow for logarithm of sigma/\lambda^2 where sigma is the perimeter and lambda is the volume. SRCF is RCF perturbated by a Brownian noise and has the remarkable property that it can be intertwined with the Brownian motion, yielding a generalization of Pitman "2M-X" theorem. We prove that along RCF, entropy Et for curvature as well as h_t:=sigma_t/lambda_t are non-increasing. We deduce infinite lifetime and convergence to a disk after normalization. For SRCF the situation is more complicated. The process (h_t)_t is always a supermartingale. For (E_t)_t to be a supermartingale, we need that the starting set is invariant by the isometry group G_n generated by the reflection with respect to the vertical line and the rotation of angle 2\pi/n with n>= 3. But for proving infinite lifetime, we need invariance of the starting set by G_n with $n>= 7. We provide the first SRCF with infinite lifetime which cannot be reduced to a finite dimensional flow. Gage inequality plays a major role in our study of the regularity of flows, as well as a careful investigation of morphological skeletons. We characterize symmetric convex sets with star shaped skeletons in terms of properties of their Gauss map. Finally, we establish a new isoperimetric estimate for these sets, of order 1/n^4 where n is the number of branches of the skeleton.