  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.
