Gaussian fluctuations for the open one-dimensional KPZ equation

In this talk we consider the open one-dimensional KPZ equation on the interval $[0,L]$ with Neumann boundary conditions. For $L \sim t^{\alpha}$ and stationary initial conditions,  we obtain matching upper and lower bounds on the variance of the height function for $\alpha \in [0,\frac23]$ for different choices of the boundary parameters. Additionally, for fixed $L$ and an arbitrary probability measure as initial conditions, we show Gaussian fluctuations for the height function as $t \to \infty$. Joint work with Sayan Das and Antonios Zitridis.