  Many models of one-dimensional local random growth are expected to lie in the
Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface
profile at advanced time may be viewed in scaled coordinates specified via
characteristic KPZ scaling exponents of one-third and two-thirds. When the long
time limit of this scaled interface is taken, it is expected -- and proved for
a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process
$\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be
embedded via the Robinson-Schensted-Knuth correspondence as the uppermost curve
in an $\mathbb{N}$-indexed system of random continuous curves, the Airy line
ensemble.
  Among our principal results is the assertion that the Airy$_2$ process enjoys
a very strong similarity to Brownian motion $B$ (of rate two) on unit-order
intervals; as a consequence, the Radon-Nikodym derivative of the law of
$\mathcal{A}$ on say $[-1,1]$, with respect to the law of $B$ on this interval,
lies in every $L^p$ space for $p \in (1,\infty)$.
  Our technique of proof harnesses a probabilistic resampling or {\em Brownian
Gibbs} property satisfied by the Airy line ensemble after parabolic shift, and
this article develops Brownian Gibbs analysis of this ensemble begun in [CH14]
and pursued in [Ham19a]. Our Brownian comparison for scaled interface profiles
is an element in the ongoing programme of studying KPZ universality via
probabilistic and geometric methods of proof, aided by limited but essential
use of integrable inputs. Indeed, the comparison result is a useful tool for
studying this universality class. We present and prove several applications,
concerning for example the structure of near ground states in Brownian last
passage percolation, or Brownian structure in scaled interface profiles that
arise from evolution from any element in a very general class of initial data.
