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Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

We study the Kardar-Parisi-Zhang equation on the half-line $x \geqslant 0$ with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter $u$ of the dynamics, and the drift $-v$ of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time $t$, it is natural to rescale parameters as $(u,v)=t^{-1/3}(a,b)$ to study the critical region. In the special case $a+b=0$, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ($a<0$) but not in the unbound phase. For instance, starting from the flat or droplet initial data, the height field near the boundary converges to the stationary process with $a>0$ and $b=0$, which is not Brownian. For $a+b\geqslant 0$, we determine exactly the large time distribution $F_{a,b}^{\rm stat}$ of the height function $h(0,t)$. As an application, we obtain the exact covariance of the height field in a half-line at two times $1\ll t_1\ll t_2$ starting from stationary initial data, as well as estimates, when starting from droplet initial data, in the limit $t_1/t_2\to 1$.