Jointly invariant measures for the Kardar-Parisi-Zhang Equation

TL;DR

This paper explicitly characterizes the invariant measures for the KPZ equation, introduces the KPZ horizon process, and explores its convergence to known models, revealing new insights into the equation's stability and measure structure.

Contribution

It provides an explicit description of the jointly invariant measures for the KPZ equation and introduces the KPZ horizon process, extending the understanding of measure invariance and stability.

Findings

Existence of a dense set of directions with discontinuous Busemann process.

Convergence of KPZ horizon to stationary horizon as temperature parameter increases.

Failure of the one force--one solution principle in certain directions.

Abstract

We give an explicit description of the jointly invariant measures for the KPZ equation. These are couplings of Brownian motions with drift, and can be extended to a process defined for all drift parameters simultaneously. We term this process the KPZ horizon (KPZH). As a corollary of this description, we resolve a recent conjecture of Janjigian, and the second and third authors by showing the existence of a random, countably infinite dense set of directions at which the Busemann process of the KPZ equation is discontinuous. This signals instability and shows the failure of the one force--one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. As the inverse temperature parameter $\beta$ for the KPZ equation goes to $\infty$, the KPZH converges to the stationary horizon (SH) first introduced by Busani, and studied further by Busani and the third and fourth authors. As $\beta \searrow 0$, the KPZH converges to a coupling of Brownian motions that differ by linear shifts, which is a jointly invariant measure for the Edwards-Wilkinson fixed point.