A flow-type scaling limit for random growth with memory

Abstract

We study a stochastic Laplacian growth model, where a set $\mathbf{U}\subseteq {\mathbb{R}}^{\mathrm{d}}$ grows according to a reflecting Brownian motion in $\mathbf{U}$ stopped at level sets of its boundary local time. We derive a scaling limit for the leading-order behavior of the growing boundary (i.e. "interface"). It is given by a geometric flow-type PDE. It is obtained by an averaging principle for the reflecting Brownian motion. We also show that this geometric flow-type PDE is locally well-posed, and its blow-up times correspond to changes in the diffeomorphism class of the growth model. Our results extend those of Dembo-Groisman-Huang-Sidoravicius '21, which restricts to star-shaped growth domains and radially outwards growth, so that in polar coordinates, the geometric flow transforms into a simple ODE with infinite lifetime. Also, we remove the "separation of scales" assumption that was taken in Dembo-Groisman-Huang-Sidoravicius '21; this forces us to understand the local geometry of the growing interface.