  We obtain several exact results for universal distributions involving the
maximum of the Airy$_2$ process minus a parabola and plus a Brownian motion,
with applications to the 1D Kardar-Parisi-Zhang (KPZ) stochastic growth
universality class. This allows to obtain (i) the universal limit, for large
time separation, of the two-time height correlation for droplet initial
conditions, e.g. $C_{\infty} = \lim_{t_2/t_1 \to +\infty} \overline{h(t_1)
h(t_2)}^c/\overline{h(t_1)^2}^c$, with $C_{\infty} \approx 0.623$, as well as
conditional moments, which quantify ergodicity breaking in the time evolution;
(ii) in the same limit, the distribution of the midpoint position $x(t_1)$ of a
directed polymer of length $t_2$, and (iii) the height distribution in
stationary KPZ with a step. These results are derived from the replica Bethe
ansatz for the KPZ continuum equation, with a "decoupling assumption" in the
large time limit. They agree and confirm, whenever they can be compared, with
(i) our recent tail results for two-time KPZ with de Nardis, checked in
experiments with Takeuchi, (ii) a recent result of Maes and Thiery on midpoint
position.
