On the two-point function of the one-dimensional KPZ equation

TL;DR

This paper applies Malliavin calculus to derive the two-point function of the slope in the 1D KPZ equation, linking it to the polymer end-point distribution and the variance derivative of the KPZ solution.

Contribution

It introduces a novel application of Malliavin calculus to analyze the two-point function of the KPZ slope, connecting it to polymer distributions and variance derivatives.

Findings

Derived the two-point function in terms of polymer end-point distribution.

Connected the distribution to the derivative of the variance of the KPZ solution.

Established a new analytical approach for KPZ correlation functions.

Abstract

In this short communication we show that basic tools from Malliavin calculus can be applied to derive the two-point function of the slope of the one-dimensional KPZ equation, starting from an arbitrary two-sided Brownian motion, in terms of the polymer end-point annealed distribution associated to the stochastic heat equation. We also prove that this distribution is given in terms of the derivative of the variance of the solution of the KPZ equation.