Integration by Parts and the KPZ Two-Point Function

TL;DR

This paper uses Malliavin calculus and Stein's method to analyze the KPZ fixed point, establishing relations between its two-point function, the location of maxima, and asymptotic independence properties.

Contribution

It introduces a novel application of integration by parts and Stein's method to study the KPZ fixed point's covariance structure and fluctuation independence.

Findings

Derived the density of the maximum location in terms of the variance's second derivative.

Established a relation between the two-point function and the maximum's location.

Proved asymptotic independence of the spatial derivative process from initial data.

Abstract

In this article we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (covariance) function of the spatial derivative process and the location of the maximum of an Airy process plus Brownian motion minus a parabola. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. In the stationary regime, we find the same density related to limit fluctuations of a second-class particle. We further develop an adaptation of Stein's method that implies asymptotic independence of the spatial derivative process from the initial data.