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We introduce a family of Markov growth processes on discrete height functionsdefined on the 2-dimensional square lattice. Each height function correspondsto a configuration of the six vertex model on the infinite square lattice. Wefocus on the stochastic six vertex model corresponding to a particulartwo-parameter family of weights within the ferroelectric regime. It is believed(and partially proven, see Aggarwal, arXiv:2004.13272) that the stochastic sixvertex model displays nontrivial pure (i.e., translation invariant and ergodic)Gibbs states of two types, KPZ and liquid. These phases have very differentlong-range correlation structure. The Markov processes we construct preservethe KPZ pure states in the full plane. We also show that the same processes puton the torus preserve arbitrary Gibbs measures for generic six vertex weights(not necessarily in the ferroelectric regime).  Our dynamics arise naturally from the Yang-Baxter equation for the six vertexmodel via its bijectivisation, a technique first used in Bufetov-Petrov(arXiv:1712.04584). The dynamics we construct are irreversible; in particular,the height function has a nonzero average drift. In each KPZ pure state, weexplicitly compute the average drift (also known as the current) as a functionof the slope. We use this to analyze the hydrodynamics of a non-stationaryversion of our process acting on quarter plane stochastic six vertexconfigurations. The fixed-time limit shapes in the quarter plane model wereobtained in Borodin-Corwin-Gorin (arXiv:1407.6729).

Irreversible Markov Dynamics and Hydrodynamics for KPZ States in the Stochastic Six Vertex Model