Stochastic dynamics of particle systems on unbounded degree graphs

TL;DR

This paper develops a mathematical framework to analyze the stochastic dynamics of an infinite particle system with unbounded interactions on graphs, proving existence and uniqueness of solutions and linking to Gibbs states.

Contribution

It introduces a novel approach combining finite volume approximation and Ovsjannikov method to handle unbounded interactions in infinite particle systems.

Findings

Proved existence and uniqueness of solutions for the SDE system.

Constructed stochastic dynamics associated with Gibbs states.

Addressed technical challenges of unbounded interaction growth.

Abstract

We consider an infinite system of coupled stochastic differential equations (SDE) describing dynamics of the following infinite particle system. Each partricle is characterised by its position $x\in \mathbb{R}^{d}$ and internal parameter (spin) $\sigma _{x}\in \mathbb{R}$. While the positions of particles form a fixed ("quenched") locally-finite set (configuration) $ \gamma \subset $ $\mathbb{R}^{d}$, the spins $\sigma _{x}$ and $\sigma _{y}$ interact via a pair potential whenever $\left\vert x-y\right\vert <\rho $, where $\rho >0$ is a fixed interaction radius. The number $n_{x}$ of particles interacting with a particle in positionn $x$ is finite but unbounded in $x$. The growth of $n_{x}$ as $x\rightarrow \infty $ creates a major technical problem for solving our SDE system. To overcome this problem, we use a finite volume approximation combined with a version of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a scale of Banach spaces of weighted sequences. As an application example, we construct stochastic dynamics associated with Gibbs states of our particle system.