Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation

Abstract

We show that the Radon-Nikodym derivative of the law of the spatial increments (with endpoints away from the origin) of inhomogeneous Brownian last passage percolation (LPP) with non-decreasing initial data against the Wiener measure $\mu$ on compacts is in $L^{\infty-}(\mu)$; and for any fixed $p>1$, the $L^p$ norm is at most of the order $O_p(\mathrm{e}^{d_pm^2\log m})$ for some $p$-dependent constant $d_p>0$. Furthermore, when the initial data is homogeneous, we establish optimal growth on $L^p$ norms ($\asymp O(\exp(dm^2))$) of the Radon-Nikodym derivative of the Brownian LPP (i.e. top line of an $m$-level Dyson Brownian motion) away from the origin, as the number of curves $m$ tends to infinity, for all $p>1$ sufficiently large. As an application of our framework, we show that the Radon-Nikodym derivative of certain toy models for the KPZ fixed point lies in $L^{\infty-}(\mu)$, inspired by its variational characterisation in terms of the directed landscape.