The stochastic renormalized mean curvature flow for planar convex sets
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ARNAUDON, Marc | |
| hal.structure.identifier | Institut Élie Cartan de Lorraine [IECL] | |
| dc.contributor.author | COULIBALY-PASQUIER, Koléhè | |
| hal.structure.identifier | Toulouse School of Economics [TSE-R] | |
| dc.contributor.author | MICLO, Laurent | |
| dc.date.accessioned | 2024-04-04T02:32:36Z | |
| dc.date.available | 2024-04-04T02:32:36Z | |
| dc.date.created | 2023-11-14 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/190398 | |
| dc.description.abstractEn | 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 $E_t$ 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. | |
| dc.description.sponsorship | Toulouse Graduate School défis en économie et sciences sociales quantitatives - ANR-17-EURE-0010 | |
| dc.language.iso | en | |
| dc.subject.en | Stochastic mean curvature evolution | |
| dc.subject.en | Morphological skeleton | |
| dc.title.en | The stochastic renormalized mean curvature flow for planar convex sets | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Probabilités [math.PR] | |
| dc.identifier.arxiv | 2303.07921 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-04018606 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-04018606v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=ARNAUDON,%20Marc&COULIBALY-PASQUIER,%20Kol%C3%A9h%C3%A8&MICLO,%20Laurent&rft.genre=preprint |
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