Fluctuation exponents of the half-space KPZ at stationarity
Yu Gu, Ran Tao
TL;DR
This work analyzes the half-space KPZ equation with a Neumann boundary in stationary Brownian initial data, recasting the problem via the Hopf-Cole transform into a half-space stochastic heat equation and a corresponding continuum directed random polymer (CDRP). A central variance identity links height fluctuations at the wall to transversal fluctuations of the half-space polymer endpoint, enabling precise fluctuation bounds across subcritical, critical, and extended critical boundary regimes, along with an explicit mean growth rate as a function of the boundary parameter. The authors establish optimal endpoint-displacement controls in the bound phase, derive symmetry reductions for u↔−u, and prove extended-critical regime bounds, clarifying depinning-like transitions and the universality of fluctuation exponents in half-space KPZ. The results hinge on a rigorous blend of Gaussian integration by parts, Feynman-Kac-type representations, and careful analysis of Robin heat kernels, Green's functions, and half-space CDRP measures. Collectively, the paper advances understanding of boundary-driven KPZ fluctuations and their phase-transition–driven scaling in half-space geometries.
Abstract
We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. Utilizing this identity, we obtain estimates for the polymer endpoints, leading to optimal fluctuation exponents for the height function in both the subcritical and critical regimes, as well as an optimal upper bound for the fluctuation exponents in the extended critical regime. We also compute the average growth rate as a function of the boundary parameter.