Integration by parts and the KPZ two-point function

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