From Particles with Random Potential to a Nonlinear Vlasov–Fokker–Planck Equation

We consider large time and infinite particle limit for a system of particles living in random potentials. The randomness enters the potential through an external ergodic Markov process, modeling oscillating environment with good statistical averaging properties. From each individual particle’s point of view, both law of large number and central limit theorem type of averaging are possible. Problems of this type have been well studied and are known as random evolutions. Instead of one particle, we focus on the collective behavior of infinite particles. We separately rescale potential functions (type one) which annihilates the equilibrium measure of the ergodic environment process, and the potential functions which may not annihilate such measure (type two). Appropriately rescaled to the macroscopic limit, type two potentials give a transport term while type one potentials give a nonlinear diffusion term. The resulting equation is a version of nonlinear Vlasov–Fokker–Planck equation. We will also prove the uniqueness of solution for such equation.