  We study ``V-shaped'' solutions to the KPZ equation, those having opposite
asymptotic slopes $\theta$ and $-\theta$, with $\theta>0$, at positive and
negative infinity, respectively. Answering a question of Janjigian,
Rassoul-Agha, and Sepp\"al\"ainen, we show that the spatial increments of
V-shaped solutions cannot be statistically stationary in time. This completes
the classification of statistically time-stationary spatial increments for the
KPZ equation by ruling out the last case left by those authors.
  To show that these V-shaped time-stationary measures do not exist, we study
the location of the corresponding ``viscous shock,'' which, roughly speaking,
is the location of the bottom of the V. We describe the limiting rescaled
fluctuations, and in particular show that the fluctuations of the shock
location are not tight, for both stationary and flat initial data. We also show
that if the KPZ equation is started with V-shaped initial data, then the
long-time limits of the time-averaged laws of the spatial increments of the
solution are mixtures of the laws of the spatial increments of $x\mapsto
B(x)+\theta x$ and $x\mapsto B(x)-\theta x$, where $B$ is a standard two-sided
Brownian motion.
