## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 21, Number (2016), paper no. 60, 52 pp.

### Invariant measures of mass migration processes

Lucie Fajfrová, Thierry Gobron, and Ellen Saada

#### 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 ($k\geq 1$) 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 zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We derive 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), condensation phenomena, and their links for MMP. We illustrate our results on many examples; we prove the coexistence of condensation and attractiveness in one of them.

#### Article information

**Source**

Electron. J. Probab. Volume 21, Number (2016), paper no. 60, 52 pp.

**Dates**

Received: 1 July 2015

Accepted: 20 July 2016

First available in Project Euclid: 30 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1475266508

**Digital Object Identifier**

doi:10.1214/16-EJP4399

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]

**Keywords**

interacting particle systems multiple jumps product invariant measures attractiveness zero-range process misanthropes process target process mass migration process condensation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Fajfrová, Lucie; Gobron, Thierry; Saada, Ellen. Invariant measures of mass migration processes. Electron. J. Probab. 21 (2016), paper no. 60, 52 pp. doi:10.1214/16-EJP4399. https://projecteuclid.org/euclid.ejp/1475266508.