Integration by parts and the KPZ two-point function. (English) Zbl 1498.60135

Summary: 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 (correlation) 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.


60F99 Limit theorems in probability theory
60H07 Stochastic calculus of variations and the Malliavin calculus
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
82D60 Statistical mechanics of polymers
60J65 Brownian motion
Full Text: DOI arXiv


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