  We consider the system of one-sided reflected Brownian motions which is in
variational duality with Brownian last passage percolation. We show that it has
integrable transition probabilities, expressed in terms of Hermite polynomials
and hitting times of exponential random walks, and that it converges in the
1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov
process defined in [MQR17] and believed to govern the long time large scale
fluctuations for all models in the KPZ universality class. Brownian last
passage percolation was shown recently in [DOV18] to converge to the Airy sheet
(or directed landscape), defined there as a strong limit of a functional of the
Airy line ensemble. This establishes the variational formula for the KPZ fixed
point in terms of the Airy sheet.
