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.
This research was supported in part by the National Council of Scientific Researches (CNPQ, Brazil) Grant 305356/2019-4, and by Foundation for Support of Research in the State of Rio de Janeiro (FAPERJ, Brazil) Grant E-26/202.636/2019.
The author would like to thank Patrik Ferrari, Jeremy Quastel for useful comments and enlightening discussions concerning this subject, and to thank Daniel Remenik for pointing out to me the differentiability of from the Fredholm determinant formula for the KPZ fixed point . Much of this work was developed during the XXIII Brazilian School of Probability, and highly inspired by the Malliavin’s Calculus classes given by D. Nualart , for which the author is very grateful.
"Integration by parts and the KPZ two-point function." Ann. Probab. 50 (5) 1755 - 1780, September 2022. https://doi.org/10.1214/22-AOP1564