Asymptotic behaviour of a tagged particle in an inhomogeneous zero-range process

We consider an infinite particle system living on a torus; from the microscopic point of view, the particles move in a lattice and their evolution is described by a spatially inhomogeneous zero-range process: each particle jumps following a symmetric random walk, with rate depending on the number of particles belonging to the same site, and weakly on the position of the site itself. In particular, we study the asymptotic behaviour of a tagged particle: under a diffusive rescaling of space and time, we prove that the process described by the tagged particle converges in law to a driftless diffusion. The proof is based on a local ergodic property and martingale convergence arguments.