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

2023/11/27 by Dembo, Amir, Yang, Kevin
#35R60 #58J65 #60H15 #82C24 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2311.16095

Abstract

We study a stochastic PDE model for an evolving set \mathbbM(t)⊆ℝ^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 ℝ^d+1, modulated by a Dirichlet-to-Neumann operator. We also show that for 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.

Related

• KPZ equation, its renormalization and invariant measures
• Convergence of the KMP model to the KPZ equation
• Integrable fluctuations in the KPZ universality class
• Hierarchy of KPZ limits arising from directed random walk models in random media
• Energy solutions of KPZ are unique