Eternal solutions of the KPZ fixed point are completely classified as patchings of Busemann functions, with shocks at patch boundaries that coalesce forward in time and can form backward.
One-sided reflected Brownian motions and the KPZ fixed point.arXiv e-prints, page arXiv:2002.02922, 2020
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Classification of the eternal solutions and multiple coalescing shocks in the KPZ fixed point
Eternal solutions of the KPZ fixed point are completely classified as patchings of Busemann functions, with shocks at patch boundaries that coalesce forward in time and can form backward.