From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

Barraquand and Le Doussal introduced a family of stationary measures for the KPZ fixed point on an interval with Neumann boundary conditions, and predicted that they arise as scaling limit of stationary measures of all models in the KPZ class on an interval. In this paper we provide a rigorous analysis of an example within the scope of this prediction: the stationary measures for the KPZ fixed point on an interval arise as the scaling limits of the height increment processes for the open asymmetric simple exclusion process in the steady state, with parameters changing appropriately as the size of the system tends to infinity