The stochastic renormalized mean curvature flow for planar convex sets

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 $\mathcal{E}_t$ 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 $(\mathcal{E}_t)_t$ 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 $n\ge 3$. But for proving infinite lifetime, we need invariance of the starting set by $G_n$ with $n\ge 7$. 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.