Spatial dynamics in interacting systems with discontinuous coefficients and their continuum limits

We consider a discrete model in which particles are characterized by two quantities X and Y ; both quantities evolve in time according to stochastic dynamics and the equation that governs the evolution of Y is also influenced by mean-field interaction between the particles. We allow for discontinuous coefficients and random initial condition and, under suitable assumptions, we prove that in the limit as the number of particles grows to infinity the dynamics of the system is described by the solution of a Fokker–Planck partial differential equation. We provide the existence and uniqueness of a solution to the latter and show that such solution arises as the limit in probability of the empirical measures of the system.