Invariant measures of Mass Migration Processes

TL;DR

This paper introduces the Mass Migration Process (MMP), a conservative particle system on integer lattices, and characterizes its invariant measures, including explicit solutions for special cases and analysis of properties like attractiveness and condensation.

Contribution

It defines the MMP, generalizes existing processes, and provides necessary and sufficient conditions for invariant measures, including explicit solutions and properties analysis.

Findings

Explicit invariant measure conditions for MMP

Full characterization of invariant measures under attractiveness

Demonstration of coexistence of condensation and attractiveness

Abstract

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.