Invariant measures and shocks in the KPZ fixed point

We construct a family of invariant measures from the perspective of a shock in the KPZ fixed point.

These measures are parameterized by a positive number $\theta > 0$, and are supported on functions $f$ satisfying $\lim_{|x| \to \infty} \frac{f(x)}{|x|} = 2\theta$.

Each can be described as the sum of a Brownian motion and an independent Bessel-$3$ process with drift.

We show that these measures appear as the $L \to \infty$ limit of the (conjectural) stationary measures for the conjectural open KPZ fixed point on $[0,L]$, after recentering by an appropriately defined shock location.

Furthermore, we show that, with respect to the standard, deterministic recentering at $x = 0$, all extremal invariant measures for the KPZ fixed point are Brownian motions with drift.

To do this, we first show that any extremal invariant measures must be supported on functions having fixed asymptotic slopes at $\pm \infty$.

Using a one-force-one-solution principle from the work of Busani, Seppäläinen, and the second author, this rules out all other invariant measures except those having left slope $-2\theta$ and right slope $+2\theta$ for some $\theta > 0$.

To handle this case, we derive the limiting fluctuations of the shock for a special choice of initial condition.

Additionally, we derive the limiting fluctuations of the shock for the case of the invariant measure from the perspective of a shock, and for the case of initial data $f(x) = 2\theta|x|$.