  The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to
admit Brownian motion with a linear drift as a stationary distribution (modulo
additive constants). We show that these solutions are attractive, a result
known as a one force--one solution (1F1S) principle or synchronization: the
solution to the KPZ equation started in the distant past from an initial
condition with a given slope will converge almost surely to a Brownian motion
with that drift, which shows in particular that these invariant measures are
totally ergodic. Our proof constructs the Busemann process for the equation,
which gives the natural jointly stationary coupling of all of these stationary
solutions. Synchronization then holds simultaneously (on a single full
probability event) for all but an at most countable random set of asymptotic
slopes. This set of exceptional slopes of instability for which synchronization
fails is either almost surely empty or almost surely dense. Along the way, we
prove a shape theorem which implies almost sure stochastic homogenization of
the KPZ equation, for which the Busemann process gives the process of
correctors. We also show that the forward and backward point-to-point and
point-to-line continuum polymers converge to semi-infinite continuum polymers
whose transitions are Doob transforms via Busemann functions of the transitions
of the finite length polymers.
