kpz-Type Equation from Growth Driven by a Non-Markovian Diffusion

Abstract

We study a stochastic pde model for an evolving set \(\mathbb {M}({t})\subseteq {\mathbb {R}}^{\textrm{d}+1}\) that resembles a continuum version of origin-excited or reinforced random walk (Benjamini and Wilson in Electron Commun Probab 8:86–92, 2003; Davis in Probab Theory Relat Fields 84(2):203–229, 1990; Kosygina and Zerner in Bull Inst Math Acad Sinica (N.S.) 8(1):105–157, 2013; Kozma in Oberwolfach Rep 27:1552, 2007; Kozma in: European congress of mathematics. European Mathematical Society, Zurich, 429–443, 2013). 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 \({\mathbb {R}}^{\textrm{d}+1}\), modulated by a Dirichlet-to-Neumann operator. We also show that, for \(\textrm{d}+1=2\), the regularization in this kpz-type equation can be removed after renormalization. To the best of our knowledge, this gives the first instance of kpz-type behavior in Laplacian growth, which investigated (for somewhat different models) in Parisi and Zheng (Phys Rev Lett 53:1791, 1984), Ramirez and Sidoravicius (J Eur Math Soc 6(3):293–334, 2004).