Stochastic dynamics of particle systems on unbounded degree graphs

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.