A flow‐type scaling limit for random growth with memory

Abstract

We study a stochastic Laplacian growth model, where a set grows according to a reflecting Brownian motion in 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.