Monge–Ampère gravitation as a Γ-limit of good rate functions

Monge-Ampère gravitation is a modification of the classical Newtonian gravitation where the linear Poisson equation is replaced by the nonlinear Monge-Ampère equation. This paper is concerned with the rigorous derivation of Monge-Ampère gravitation for a finite number of particles from the stochastic model of a Brownian point cloud, following the formal ideas of a recent work by the third author (Bull. Inst. Math. Acad. Sin. 11:1(2016), 23-41). This is done in two steps. First, we compute the good rate function corresponding to a large deviation problem related to the Brownian point cloud at fixed positive diffusivity. Second, we study the Γ−convergence of this good rate function, as the diffusivity tends to zero, toward a (non-smooth) Lagrangian encoding the Monge-Ampère dynamic. Surprisingly, the singularities of the limiting Lagrangian correspond to dissipative phenomena. As an illustration, we show that they lead to sticky collisions in one space dimension.