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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.