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

We study a stochastic PDE model for an evolving set $\mathbb{M}(t)\subseteq\mathbb{R}^{\mathrm{d}+1}$ that resembles a continuum version of origin-excited or reinforced random walk. We show that long-time fluctuations of an associated height function are given by a regularized Kardar-Parisi-Zhang (KPZ)-type PDE on a hypersurface in $\mathbb{R}^{\mathrm{d}+1}$, modulated by a Dirichlet-to-Neumann operator. We also show that for $\mathrm{d}+1=2$, the regularization in this KPZ-type equation can be removed after renormalization. To our knowledge, this gives the first instance of KPZ-type behavior in Laplacian growth, which was asked about (for somewhat different models) in Parisi-Zhang '84 and Ramirez-Sidoravicius '04.