  The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process, recently
introduced by Matetski, Quastel, Remenik (arXiv:1701.00018), that describes the
limit fluctuations of the height function associated to the totally asymmetric
simple exclusion process (TASEP), and it is conjectured to be at the centre of
the KPZ universality class. Our main result is that the KPZ incremental process
converges weakly to its invariant measure, given by a two-sided Brownian motion
with zero drift and diffusion coefficient 2. The heart of the proof is the
coupling method that allows us to compare the TASEP height function with its
invariant process, which under the KPZ scaling turns into uniform estimates for
the KPZ fixed point.
