Stochastic dynamics on infinite product manifolds:twenty five years after

We consider an infinite system of stochastic differential equations in a compact manifold M. The equations are labeled by vertices of a geometric graph with unbounded vertex degrees and coupled via nearest neighbour interaction. We prove the global existence and uniqueness of strong solutions and construct in this way stochastic dynamics associated with Gibbs measures describing equilibrium states of a (quenched) system of particles with positions forming a typical realization of a Poisson or Gibbs point process in Rd.