We show that the increments of the KPZ fixed point started from arbitrary initial data are 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 by Sarkar and Virag in 2021. As applications, we investigate geometric properties of the graph of the KPZ fixed point, obtaining a characterisation for the positivity of certain hitting probabilities thereof using a certain thermal capacity and compute essential suprema of Hausdorff dimensions of these random intersections.

The arguments above can be extended to 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. This allows us to 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.