Brownian Aspects of the KPZ Fixed Point

TL;DR

This paper investigates the Brownian limiting behavior of the KPZ fixed point, focusing on local space regularity and long-term evolution, using a variational approach to analyze different initial conditions.

Contribution

It introduces a variational method to study the KPZ fixed point's Brownian aspects, enabling analysis from various initial data without relying solely on explicit formulas.

Findings

Confirmed Brownian local space regularity

Established Brownian-like long-time evolution

Applied variational approach for direct proofs

Abstract

The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process that is conjectured to be at the core of the KPZ universality class. In this article we study two aspects the KPZ fixed point that share the same Brownian limiting behaviour: the local space regularity and the long time evolution. Most of the results that we will present here were obtained by either applying explicit formulas for the transition probabilities or applying the coupling method to discrete approximations. Instead we will use the variational description of the KPZ fixed point, allowing us the possibility of running the process starting from different initial data (basic coupling), to prove directly the aforementioned limiting behaviours.