  We introduce a family of Markov growth processes on discrete height functions
defined on the 2-dimensional square lattice. Each height function corresponds
to a configuration of the six vertex model on the infinite square lattice. We
focus on the stochastic six vertex model corresponding to a particular
two-parameter family of weights within the ferroelectric regime. It is believed
(and partially proven, see Aggarwal, arXiv:2004.13272) that the stochastic six
vertex model displays nontrivial pure (i.e., translation invariant and ergodic)
Gibbs states of two types, KPZ and liquid. These phases have very different
long-range correlation structure. The Markov processes we construct preserve
the KPZ pure states in the full plane. We also show that the same processes put
on 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 vertex
model 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, we
explicitly compute the average drift (also known as the current) as a function
of the slope. We use this to analyze the hydrodynamics of a non-stationary
version of our process acting on quarter plane stochastic six vertex
configurations. The fixed-time limit shapes in the quarter plane model were
obtained in Borodin-Corwin-Gorin (arXiv:1407.6729).
