  For each $t\geq 1$ we construct an $\mathbf{N}$-indexed ensemble of random
continuous curves with three properties: 1. The lowest indexed curve is
distributed as the time $t$ Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ)
stochastic PDE with narrow wedge initial data; 2. The entire ensemble satisfies
a resampling invariance which we call the $\mathbf{H}$-Brownian Gibbs property
(with $\mathbf{H}(x)=e^{x}$); 3. Increments of the lowest indexed curve, when
centered by $-t/24$ and scaled down vertically by $t^{1/3}$ and horizontally by
$t^{2/3}$, remain uniformly absolutely continuous (i.e. have tight
Radon-Nikodym derivatives) with respect to Brownian bridges as time $t$ goes to
infinity.
  This construction uses as inputs the diffusion that O'Connell discovered in
relation to the O'Connell-Yor semi-discrete Brownian polymer, the convergence
result of Nica of the lowest indexed curve of that diffusion to the solution of
the KPZ equation with narrow wedge initial data, and the one-point distribution
formula proved by Amir-Corwin-Quastel for the solution of the KPZ equation with
narrow wedge initial data.
  We provide four main applications of this construction: 1. Uniform (as $t$
goes to infinity) Brownian absolute continuity of the time $t$ solution to the
KPZ equation with narrow wedge initial data, even when scaled vertically by
$t^{1/3}$ and horizontally by $t^{2/3}$; 2. Universality of the $t^{1/3}$
one-point (vertical) fluctuation scale for the solution of the KPZ equation
with general initial data; 3. Concentration in the $t^{2/3}$ scale for the
endpoint of the continuum directed random polymer; 4.Exponential upper and
lower tail bounds for the solution at fixed time of the KPZ equation with
general initial data.
