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 Brenier (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
-convergence
of this good rate function, as the diffusivity tends to zero, toward a (nonsmooth)
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.
Keywords
Monge–Ampère gravitation, large deviations,
$\Gamma$-convergence, Lagrangian mechanics, interacting
particle systems
