Matching Large Deviation Bounds of the Zero-Range Process in the whole space

TL;DR

This paper establishes matching large deviation bounds for the zero-range process on the entire space, extending previous results and providing a comprehensive probabilistic and analytical framework.

Contribution

It provides the first complete large deviation bounds for the zero-range process in all dimensions, removing previous restrictions and extending key estimates and equations.

Findings

Matching upper and lower large deviation bounds are obtained.

Superexponential concentration on paths with finite entropy dissipation is proved.

The theory of the parabolic-hyperbolic skeleton equation is extended to the whole space.

Abstract

We consider the large deviations of the hydrodynamic rescaling of the zero-range process on $\mathbb{Z}^d$ in any dimension $d\ge 1$. Under mild and canonical hypotheses on the local jump rate, we obtain matching upper and lower bounds, thus resolving the problem opened by \cite{KL99}. On the probabilistic side, we extend the superexponential estimate to any dimension, and prove the superexponential concentration on paths with finite entropy dissipation. In addition, we extend the theory of the parabolic-hyperbolic skeleton equation to the whole space, and remove global convexity/concavity assumptions on the nonlinearity.