Stochastic bounds on the zero range processes with superlinear jump rates

We present some results concerning the regularity of dynamics of the zero-range process. These results are achieved by using coupling based on attractivity of the models. The essential novelty is that these methods are carried out without any restriction on the increase of the jump rates depending on the local configurations, in contrast to the usually required sub-linear growth conditions. Based on the bounds presented here, the dynamics of these models may be constructed.