The KPZ fixed point and Brownian motion share the same null sets

TL;DR

This paper demonstrates that the increments of the KPZ fixed point are mutually absolutely continuous with Brownian motion on compact sets, and explores the relationships and properties of the KPZ fixed point, Airy sheet, and Brownian motion.

Contribution

It extends the absolute continuity relations of the KPZ fixed point with Brownian motion and analyzes the Hausdorff dimensions and hitting probabilities of these stochastic processes.

Findings

KPZ fixed point increments are mutually absolutely continuous with Brownian motion on compacts.

Additive Brownian motion is absolutely continuous with respect to the centred Airy sheet on compacts.

The paper characterizes hitting probabilities and Hausdorff dimensions of the KPZ fixed point and related processes.

Abstract

We show that the increments of the KPZ fixed point started from arbitrary initial data are \emph{mutually} absolutely continuous with respect to Brownian motion with diffusion parameter $2$ on compacts, extending the one-sided Brownian absolute continuity relation of the KPZ fixed point established in \cite{sarkar2021brownian}. We also show that additive Brownian motion is absolutely continuous with respect to the centred Airy sheet on compacts, but it is not mutually absolutely continuous globally. As applications, we show that with probability strictly between zero and one, there exist record times of the KPZ fixed point away from any reference point, obtain a characterisation for the hitting probabilities of the graph of the KPZ fixed point to be positive in terms of a certain thermal capacity in the sense of \cite{watson1978corrigendum, watson1978thermal} and compute essential suprema of Hausdorff dimensions of these random intersections. Finally, we compute essential suprema of Hausdorff dimensions of images of subsets in the plane under the Airy sheet and give a condition for the positivity of their Lebesgue measure in terms of Bessel-Riesz capacity.