KPZ-type equation from growth driven by a non-Markovian diffusion

Abstract: 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.