  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.
