Viscous shock fluctuations in KPZ

TL;DR

This work resolves the existence of time-stationary V-shaped solutions in the KPZ equation by showing that no invariant measure can support a V-shaped profile with θ>0. The authors connect this nonexistence to the fluctuations of the viscous shock location b_t and provide precise asymptotics for the shock in both stationary and flat-initial-data regimes, including Gaussian and GOE-based limits. A central theme is the detailed analysis of increments and their stationarity properties via jointly invariant measures ν_θ and tilted measures ḥν_θ, together with the KPZ fixed point framework and half-space polymer techniques. The results yield a complete picture: V-shaped initial data lead to time-averaged increment laws that are mixtures of the two drifted Brownian-increment laws, with symmetric basins yielding an equal mixture, thereby extending connections with ASEP and clarifying basins of attraction for KPZ dynamics.

Abstract

We study ``V-shaped'' solutions to the KPZ equation, those having opposite asymptotic slopes $θ$ and $-θ$, with $θ>0$, at positive and negative infinity, respectively. Answering a question of Janjigian, Rassoul-Agha, and Seppäläinen, we show that the spatial increments of V-shaped solutions cannot be statistically stationary in time. This completes the classification of statistically time-stationary spatial increments for the KPZ equation by ruling out the last case left by those authors. To show that these V-shaped time-stationary measures do not exist, we study the location of the corresponding ``viscous shock,'' which, roughly speaking, is the location of the bottom of the V. We describe the limiting rescaled fluctuations, and in particular show that the fluctuations of the shock location are not tight, for both stationary and flat initial data. We also show that if the KPZ equation is started with V-shaped initial data, then the long-time limits of the time-averaged laws of the spatial increments of the solution are mixtures of the laws of the spatial increments of $x\mapsto B(x)+θx$ and $x\mapsto B(x)-θx$, where $B$ is a standard two-sided Brownian motion.

Theorems & Definitions (71)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10