  We study a stochastic geometric flow that describes a growing submanifold
$\mathbb{M}(t)\subseteq\mathbb{R}^{\mathrm{d}+1}$. It is an SPDE that comes
from a continuum version of origin-excited random walk or once-reinforced
random walk. It is given by simultaneously smoothing and inflating the boundary
of $\mathbb{M}(t)$ in a neighborhood of the boundary trace of a reflecting
Brownian motion. We show that the large-scale fluctuations of an associated
height function are given by a regularized Kardar-Parisi-Zhang (KPZ)-type
equation on a manifold in $\mathbb{R}^{\mathrm{d}+1}$, modulated by a
Dirichlet-to-Neumann operator. This is shown in any dimension
$\mathrm{d}\geq1$. We also prove that in dimension $\mathrm{d}+1=2$, the
regularization in this KPZ-type SPDE can be removed after renormalization.
Thus, in dimension $\mathrm{d}+1=2$, fluctuations of the geometric flow have a
double-scaling limit given by a singular KPZ-type equation. To our knowledge,
this is the first instance of KPZ-type behavior in stochastic Laplacian growth
models, which was asked about (for somewhat different models) in Parisi-Zhang
'84 and Ramirez-Sidoravicius '04.
