A flow-type scaling limit for random growth with memory

Abstract

We study a stochastic Laplacian growth model, where a set $U\subseteq R^d$ grows according to a reflecting Brownian motion in $U$ 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 et al., 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 et al.; this forces us to understand the local geometry of the growing interface.