Invariant measures of mass migration processes

Abstract

We introduce the Mass Migration Process (MMP), a conservative particle system on N Z d . It consists in jumps of k particles (k ≥ 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.

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

“…In the same spirit of point (d), the proof in [22] implies that F(η i , η i+1 ) has to be of the form…”

Metastable Markov chains

“…Static aspects of condensation for this zero-range process and other dynamics have been examined by Jeon, March and Pittel [79], Grosskinsky, Schütz and Spohn [73], Armendáriz and Loulakis [5,6], Chleboun and Grosskinsky [41,42,43], Grosskinsky, Redig and Vafayi [71], Godrèche and Luck [69], Armendáriz, Grosskinsky and Loulakis [3], Fajfrová, Gobron and Saada [59].…”

Hydrodynamic limits for long-range asymmetric interacting particle systems

“…The last condition (iii) is a sufficient ingredient to construct the process, and forces g to be bounded if h is nontrivial (cf. equation (7.1) in [17]). However, it is not a necessary condition, and will not be used in the main body of the paper.…”

“…for 0 ≤ i, j ≤ M 0 if M 0 < ∞ and i, j ≥ 0 otherwise, which we will also assume (cf. [17]). In the case h is nontrivial, this implies a linear relation between g and h.…”

“…t starting from the product measure µ N , for which the relative entropy and localized Dirichlet form of dµ N,R t /dν ρ * satisfy (4.6) and (4.7). In particular, by the construction (cf [17]…”