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

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