We consider zero-range processes in Zd with site dependent jump
rates. The rate for a particle jump from site x to y in Zd is
given by λx​g(k)p(y−x), where p(⋅) is a probability in
Zd, g(k) is a bounded nondecreasing function of the number k
of particles in x and λ={λx​} is a collection of i.i.d.
random variables with values in (c,1], for some c>0. For almost every
realization of the environment λ the zero-range process has product
invariant measures {νλ,v​:0≤v≤c} parametrized by v,
the average total jump rate from any given site. The density of a measure,
defined by the asymptotic average number of particles per site, is an
increasing function of v. There exists a product invariant measure νλ,c​, with maximal density. Let μ be a probability measure
concentrating mass on configurations whose number of particles at site x
grows less than exponentially with ∥x∥. Denoting by Sλ​(t) the
semigroup of the process, we prove that all weak limits of {μSλ​(t),t≥0} as t→∞ are dominated, in the natural partial
order, by νλ,c​. In particular, if μ dominates νλ,c​, then μSλ​(t) converges to νλ,c​.
The result is particularly striking when the maximal density is finite and the
initial measure has a density above the maximal.Comment: Published at http://dx.doi.org/10.1214/074921707000000300 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org