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

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