Abstract
We provide two methods to construct zero-range processes with superlinear rates on Zd. In the first method these rates can grow very fast, if either the dynamics and the initial distribution are translation invariant or if only nearest neighbour translation invariant jumps are permitted, in the one-dimensional lattice. In the second method the rates cannot grow as fast but more general dynamics are allowed.
Highlights
The zero-range process was introduced by Spitzer [10] as a Markov process on NS0, where N0 is the set of non-negative integers, and S is a denumerable set
Balázs, Rassoul-Agha, Seppäläinen and Sethuraman [3] construct the zero-range process with totally asymmetric dynamics p(x, y) = 1(y − x = 1) and nearest neighbour jumps in the one-dimensional lattice Z, under the assumption that the jump rates are non-decreasing and grow at most exponentially
In this article we introduce two methods to construct zero-range processes with superlinear rates on integer lattices, and identify the associated martingales
Summary
The zero-range process was introduced by Spitzer [10] as a Markov process on NS0 , where N0 is the set of non-negative integers, and S is a denumerable set. Balázs, Rassoul-Agha, Seppäläinen and Sethuraman [3] construct the zero-range process with totally asymmetric dynamics p(x, y) = 1(y − x = 1) and nearest neighbour jumps in the one-dimensional lattice Z, under the assumption that the jump rates are non-decreasing and grow at most exponentially. Under these conditions, they prove that the process is Markov and admits a one parameter family of extremal invariant measures. The second method can be applied to quite general random walks on Zd, but is more restrictive on the rate functions
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