Killing of Transport in Lattices driven by Local Quantum Stochastic Dynamics

Systems with local dynamics are characterized by a finite velocity of propagation of perturbations, known as the Lieb-Robinson velocity. On the other hand, irreducible stochastic processes drive states towards some unique fixed point. However, combining both effects is mathematically challenging. The bounds on propagation do not depend on system-size, while the theory of mixing is mostly based on extensive upper-bounds. In a previous paper, a class of local Lindbladian operators on arbitrary lattice was constructed, for which the two effects could be combined. In this paper, we show that for some local dynamics, local observables are propagated inside a much smaller convex subset of the total Banach space. This allows us to show localization for such dynamics.