We prove that the Kipnis-Marchioro-Presutti (KMP) process converges to the Kardar-Parisi-Zhang (KPZ) equation, as time $t$ goes to infinity, in a properly scaled observation window shifted by $t^{ 3/4}$. Our proof is based on identifying the KMP process with a stochastic flow of kernels describing transition probabilities in a certain model of random walk in space-time random environment. This allows to apply a recent result of arXiv:2401.06073 proving convergence of the density field of random walks in random environment to the KPZ equation in a suitably general sense.
Categories: math.PR cond-mat.stat-mech math-ph math.MP