Small Noise Limit and Convexity for Generalized Incompressible Flows, Schrödinger Problems, and Optimal Transport

Abstract

This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schrödinger problem, Brenier's relaxed model for incompressible fluids, the so-called Brödinger problem recently introduced by M. Arnaudon & al. [3], the multiphase Brenier model, and the multiphase Brödinger problem. All of them involve the minimization of a kinetic action and/or a relative entropy of some path measures with respect to the reversible Brownian motion. As the viscosity parameter ν → 0 we establish Gammaconvergence relations between the corresponding problems, and prove the convergence of the associated pressures arising from the incompressibility constraints. We also present new results on the time-convexity of the entropy for some of the dynamical interpolations. Along the way we extend previous results by H. Lavenant [30] and J-D. Benamou & al. [10].

New estimates on the regularity of the pressure in density‐constrained mean field games

“…Yet, the sharp regularity of the pressure is in such a setting an open problem, as semi‐concavity is a reasonable conjecture advanced by Brenier, but the current achievements do not go beyond the result mentioned above. Yet, due to the multiphasic nature of the problem formulated by Brenier, it is in general not possible to translate all the available techniques into such a more complicated setting (see, for instance, where the time‐convexity of the entropy is proven, but, differently from , the same cannot be obtained for other internal energies; analogously, the same results of are also recovered in , and the same algebraic obstruction prevents from generalizing the result to more general energies). On the other hand, the works on density‐constrained MFG (including a first attempt, with a non‐variational model, in ) were inspired by previous works of the second author on crowd motion formulated as a gradient flow with density constraints (see and ), and the present technique seems possible to be applied to such a first‐order (in time) setting.…”

On the Existence of a Scalar Pressure Field in the Brödinger Problem

“…As far as we know, this work and [9] are the first articles dealing with this optimization problem (even though in [9], it is formulated using the notion of transport plans). A consequence of the analysis made in [9] (see Theorem A.1) is that the existence of solution holds for this problem if and only if the initial and finite total entropies are finite, namely if and only if for m-almost all i, ρ i 0 and ρ i 1 have densities with respect to Leb, and:…”

“…Theorem 5. Take (I, m) a probability space, ν > 0, ρ 0 and ρ 1 satisfying (7) (9). Let (ρ, c) = (ρ i , c i ) i∈I be the solution of MBrö ν (ρ 0 , ρ 1 ).…”

“…We fix (I, m) a probability space of labels for the different phases and ν > 0 a level of noise. We also fix ρ 0 = (ρ i 0 ) i∈I and ρ 1 = (ρ i 1 ) i∈I two measurable families of probability measures on T d satisfying the incompressibility condition (7) and condition (9), so that the problem MBrö ν (ρ 0 , ρ 1 ) admits a unique solution.…”

“…In fact, it is always the case, because we could show that these quantities are controlled by Hν (ρ, c), see[9, Remark A.3].…”