# Zero Range Process: Construction of Feller process by the generator

A Zero Range Process is the Feller process with generator

$Af(\eta)=\sum_{x,y} p(y-x)f(\eta(x)) (f(\eta^{x,y}-f(\eta))$

where $\eta\in \mathbb{N}_0^S$ denotes a configuration of particles at sites $S$, $p=(p(x,y)_{x,y})$ is doubly-stochastic translation invariant and $g$ some nondecreasing function. It describes the dynamics of moving particles for a given number of particles on a countable number of sites, where the jump rate depends only on the number of particles at a particular site (hence zero range). We assume first $S$ is finite.

The question arises what kind of conditions have to be imposed to guarantee existence of a Feller process with the given generator. I often see $g^*:=\sup_k |g(k+1)-g(k)|<\infty$, but I fail to see how this condition could be helpful in the case of finite site space $S$ (maybe necessary for the hydrodynamic limit?), as in that case I assume you would simply apply the state space $\mathcal{S}_K:=\{ \eta: \sum_x \eta_x =K\}$ for some $K$ and then end up with a bounded operator. A state space of the form $\mathcal{S}_{fin}\{ \eta: \sum_x \eta_x <\infty\}$ just decomposes into irreducibility classes of the above form.

To clarify my inquiry: The condition on $g^*$ above pops up everywhere. For the state space $\mathcal{S}_K$, the condition is superfluous for establishing existence of a Feller process with the generator given above, as the generator is bounded in that case. I don't see how the condition would help in the case where the state space is $\mathcal{S}_{fin}$

To give some background on used sources: I'm referring mostly to the books of Kipnis/Landim respectivley Liggett on Interacting Particle Systems, as well as the paper "An infinte particle system with zero range interactions" by Liggett, 1973.