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Barraquand and Le~Doussal introduced a family of stationary measures for the(conjectural) KPZ fixed point on an interval with Neumann boundary conditions,and predicted that they arise as scaling limit of stationary measures of allmodels in the KPZ universality class on an interval. In this paper, we showthat the stationary measures for KPZ fixed point on an interval arise as thescaling limits of the height increment processes for the open asymmetric simpleexclusion process in the steady state, with parameters changing appropriatelyas the size of the system tends to infinity.

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