Interacting particle systems with continuous spins

TL;DR

This paper develops a mathematical framework for analyzing a broad class of interacting particle systems with continuous spins, establishing existence, uniqueness, and convergence properties of their dynamics.

Contribution

It introduces a new construction of Markovian dynamics via stochastic differential equations and provides conditions for invariant measures and convergence in the subcritical regime.

Findings

Existence and uniqueness of invariant measures under certain conditions

Convergence of transition probabilities in Wasserstein-1-distance

Spatial spread is at most linear in time for sublinear drifts

Abstract

We study a general class of interacting particle systems over a countable state space $V$ where on each site $x \in V$ the particle mass $\eta(x) \geq 0$ follows a stochastic differential equation. We construct the corresponding Markovian dynamics in terms of strong solutions to an infinite coupled system of stochastic differential equations and prove a comparison principle with respect to the initial configuration as well as the drift of the process. Using this comparison principle, we provide sufficient conditions for the existence and uniqueness of an invariant measure in the subcritical regime and prove convergence of the transition probabilities in the Wasserstein-1-distance. Finally, for sublinear drifts, we establish a linear growth theorem showing that the spatial spread is at most linear in time. Our results cover a large class of finite and infinite branching particle systems with interactions among different sites.