  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.
