Optimal transport with discrete long range mean field interactions

We study an optimal transport problem where, at some intermediate time, the mass is accelerated by either an external force field, or self-interacting. We obtain regularity of the velocity potential, intermediate density, and optimal transport map, under conditions on the interaction potential that are related to the so-called Ma-Trudinger-Wang condition from optimal transport.


Motivations
The optimal transport problem goes back to a cost-minimization problem in civil engineering proposed by Monge [1], later generalized to a class of optimization problems by Kantorovich [2,3], with an elegant economic interpretation. Later, through The first equation is the conservation of mass, the second equation states that the acceleration field is given by −∇p, and the third equation is the gravitational Poisson coupling.
The reconstruction problem is to find a solution to (1.1) satisfying i.e. given the initial and final densities, as opposed to the Cauchy (or initial-value) problem, where one is given the initial density and velocity. In [9] (see also [4] for the case of the incompressible Euler equations), the reconstruction problem was formulated as a minimization problem, minimizing the action of the Lagrangian, which is a convex functional of properly chosen variables. Through this variational formulation, the reconstruction problem becomes very similar to the time-continuous formulation of the optimal transport problem of Benamou and Brenier [5]. Moreover, through the study of a dual problem, reminiscent of Monge-Kantorovich duality, partial regularity results for the velocity and the density were obtained.
The optimal transport problem of [9] was formulated as finding minimizers of the action over all ρ, p and v satisfying ∂ t ρ + ∇ · (ρv) = 0, ρ(0) = ρ 0 , ρ(T ) = ρ T , where T d denotes the d-dimensional torus, as the study in [9] was performed in the space-periodic case.

Goal of the paper
In this paper we address the problem of finding minimizers for the action for a certain class of internal energy F , and we will obtain some partial results in that direction. Apart from the application in cosmology, several authors have looked at continuous optimal transport with or without interaction, notably through their natural connections with mean-field games. We refer the reader to the work [14] and the references therein, where the connection is explored. Again, we also refer to the book [11] where the notion of gradient flows on space of measures and its relation to optimal transport are addressed. J. Liu & G. Loeper This work is about the study of regularity of minimizers to (1.3). In [10], Lee and McCann address the case where which is obviously linear in ρ. This Lagrangian corresponds to the case of a continuum of matter evolving in an external acceleration field given by ∇Q(t, x). We call this the noninteracting case for obvious reasons. This can be recast as an optimal transport problem where the cost function is given by where γ is a smooth curve connecting x and y. By assuming that Q(t, x) = εV (x) for some V satisfying the structure condition for a constant C > 0, for all (x, v) in the tangent bundle T T d , and for all unit tangent vectors u, w in the tangent space T x T d that are orthogonal to each other, Lee and McCann obtain that for a small enough ε > 0, the cost c satisfies the conditions found in [15] to ensure the regularity of the optimal map. (Note that in order to be consistent with our notations, here we changed the sign of the potential actually considered in [10].)

The noninteracting, discrete case
In this paper we will restrict ourselves to the case where the force field only acts at a single discrete time between 0 and T : The minimization problem therefore becomes for some potential Q. This restriction will allow us to remove the smallness condition on Q.

The mean-field case
We will be able to extend our result to the nonlinear situation where the force field is given by Optimal transport with discrete long-range mean-field interactions equal to ∇κ(x − y). In this case we will show that the action to minimize becomes Reasoning formally, one sees straightaway that on [0, T/2] we are solving the usual optimal transport problem in its "Benamou-Brenier" formulation [5], as well as on [T /2, T ], and therefore particles will travel with constant velocity in those two intervals. At t = T /2, the velocity v will be discontinuous. We will give a sufficient condition on κ to ensure a smooth transport map and intermediate density.
Unfortunately, the gravitational case, which corresponds to the Coulomb kernel does not satisfy our condition.

The time discretization
A natural approximation of the time-continuous problem is to partially discretize in time (1.3) into for t i = iT N , i = 0, . . . , N, a discretization of the time interval [0, T ]. Then, on each subinterval [t i−1 , t i+1 ], having set ρ(t i−1 ) and ρ(t i+1 ), one has to solve a problem of the form (1.5). For this reason, the study of the problem (1.5) seems a natural starting point. Our results show regularity on ρ(T /2) assuming ρ(0) and ρ(T ) are regular [loosely speaking, ρ(T /2) has the same regularity as ρ(0) and ρ(T )]. It would be a natural extension of our work to deal with multiple time steps, however we do not see a straightforward way to tackle this problem.

Organization of the paper
The paper is organized as follows: In Sec. 2 we state the problem formally, our main assumptions, and our results. In Sec. 3, we derive Eq. (2.17) formally by straightforward computations. In Sec. 4, we introduce a two-step optimal transport problem and prove Theorem 2.1. By assuming the conditions (H0)-(H1) we have the existence and uniqueness of the velocity potential φ. Moreover, we provide an interpretation of the cost function from a natural mechanical point of view. In Sec. 5, we introduce the condition (H2), which is crucial in obtaining the regularity of φ. In Sec. 6, upon formulating our reconstruction problem into an optimal transport problem, we have the regularity of φ and conclude Theorem 2.2. In Sec. 7, we consider the mean-field case under the condition (H2c), which is preserved by convex combinations, and then prove Theorems 2.4 and 2.5. Problem 1. We consider P 2 (R d ) (in short P 2 ) the set of probability measures on R d with finite second moment, and a functional F : and the constraints
Optimal transport with discrete long-range mean-field interactions Problem 3 is to minimize K among all ρ ∈ P 2 . It coincides with the notion of Wasserstein Barycenters, see [16,17], when From the classical results of optimal transport (see [12,13,11]), in the case where F (ρ) is convex and lower-semi-continuous (l.s.c.) in ρ there holds the following proposition.
Assume that F (ρ) is convex and l.s.c. in ρ, and that Problem 1, 2, or 3 has at least one admissible solution, then Problems 1-3 are equivalent, and moreover there holds where v, γ are respectively from Problems 1 and 2, Φ * is a convex potential such that ∇Φ * # ρ 0 = ρ, and ρ is the optimizer in Problem 3.
Proposition 2.2 gives the Euler-Lagrange equation characterizing the optimizer. It is based on the Riemannian metric induced on P 2 by W 2 (see again the above references for a complete coverage).

Proposition 2.2.
Let ρ be the optimizer in Problem 3. There exists a vector field w = ∇Ξ ∈ L 2 (dρ) and two convex potentials Φ, Ψ such that b moreover, ∇Ξ can be identified to be the gradient of F with respect to the Wasserstein metric, i.e. for all curves ρ t ⊂ P 2 ∩ Dom(F ) passing through ρ at t = 0, and such that We next characterize Ξ in the two cases of interest for this paper.
b Note that Φ * above is the Legendre transform of Φ.
We can also completely characterize the optimal velocity: With φ as in Proposition 2.1, we have the following: . Finally, we characterize the optimal map: m(x) such that m(γ(0, x)) = γ(T, x) (for γ the optimizer in Problem 2) will be given by

Assumptions
From the previous observations, and in order to motivate our assumptions, let us derive formally the equation giving the initial velocity potential φ. Let the initial density ρ 0 be supported on a bounded domain Ω 0 ⊂ R d , and the final density ρ T be supported on a bounded domain Ω T ⊂ R d , satisfying the balance condition Starting from the definition (2.13), we introduce the modified potential functions By computing the determinant of the Jacobian Dm and noting that m # ρ 0 = ρ T , i.e. that m pushes forward the measure ρ 0 onto the measure ρ T , one can derive the Monge-Ampère-type equation (see Sec. 3 for detailed computation) Optimal transport with discrete long-range mean-field interactions To ensure the regularity of the solutionφ (equivalently that of φ) to the boundary-value problem (2.17) and (2.18), it is necessary to impose certain conditions on the potential functionQ (equivalently on Q) and the domains Ω 0 , Ω T . In this paper we assumeQ satisfies the following conditions: (H0) The potential functionQ belongs to C 4 (R d ).
Note that the conditions (H0) and (H1) imply that the inverse matrix (D 2Q ) −1 exists, and ensure that Eq. (2.17) is well defined. Condition (H2) is an analog of the Ma-Trudinger-Wang (MTW) condition in optimal transportation, which is necessary for regularity results (note the factor 2 however). We shall give more detailed interpretations and examples in Sec. 5.

Results
Our first main result is the following.
Theorem 2.1. Under the assumptions (H0) and (H1) Problems 1-3 are equivalent to solving an optimal transport problem with cost function c(x, y) =Q * (x + y), whereQ * is the Legendre transform ofQ given in (2.16). There exists a potential φ such that the optimal transport map of ρ 0 onto ρ T with cost c will be given by T (x) = y such that D x Q * (x + y) = D xφ (x). Then φ = 2 T (φ − |x| 2 /2) will be the initial velocity potential as in Proposition 2.1.
Our next result is a regularity result.
As a byproduct of those two results we obtain the following: Consider the optimal transport problem with cost c(x, y) = R(x+y) for some R : R d → R convex. Then this problem is equivalent to the minimization problem (1.5) with potential Q(z) = 2 T (R * (z) − |z| 2 ), for R * being the Legendre-Fenchel transform of R.
For the mean-field case, we have the following existence and uniqueness result.
There exists a minimizer to problem (1.7). Moreover, once ρ(T /2) is known, lettingρ = ρ(T /2), the minimizer will be the same as the solution of non-interacting problem (1.5) where Q is given by (2.20) Under additional assumptions on the kernel κ and the domains, we have the following regularity result.

21)
and where b 0 , b 1 are two constants, and M is the total mass in (2.15).

Optimal transport with discrete long-range mean-field interactions
The proof of Theorem 2.5 relies on the observation that the q-convexity and the condition (H2c) are preserved under convex combinations, and therefore by convolution with the density ρ(T /2), and on some a priori C 1 estimates on the potential. We remark that the condition (H2c) is related to the condition (B4) in [21] where the cost function satisfies the MTW condition without the orthogonal restriction.

Formal Derivation of Eq. (2.17)
Throughout the following context, unless mentioned otherwise, the function φ always denotes the initial velocity potential, namely at time t = 0, the velocity In order to derive the equation for φ, let us track a single point x ∈ Ω 0 . Recalling that at t = T /2, the potential ∇Q affects the velocity v = ∇φ, the final point y = m(x) is given by The Jacobian matrix of m is where I is the d× d identity matrix, and the matrix (D 2 Q) is taken at (x+ T /2∇φ). Define and assume that the matrix (I + A) is invertible. Then Computing the determinant of Dm, we have

J. Liu & G. Loeper
Recall that the modified velocity potentialφ and potential functionQ are given byφ we have D 2Q = (I + A) and Therefore, we obtain the Monge-Ampère equation with an associated natural boundary condition Remark 3.1. In the continuous case (1.1), Loeper obtained in [9] partial regularity for φ, that holds only in the interior (with respect to time) of the domain. In particular, there was no result regarding the initial velocity. In this paper, we obtain the regularity of φ over the whole domain in the discrete case by using the regularity in optimal transportation.
We need to introduce a notion of weak solutions for Eq. (3.4).

Definition 3.1.
A functionφ is said to be a weak Brenier solution to (3.4) whenever m defined fromφ as in (3.2) is such that namely, for all B ⊂ R d Borel, there holds ρ 0 (m −1 (B)) = ρ T (B) (this is also mentioned as m pushes forward ρ 0 onto ρ T ).

Remark 3.2.
It is known that, depending on the geometry of the support of ρ T , the notion of Brenier solution might be equivalent to the notion of Aleksandrov solution.

A Two-Step Transport
Problem (1.5) falls into the class of optimal transport problems with a general cost function Optimal transport with discrete long-range mean-field interactions where the infimum is taken over all smooth curves γ(·) satisfying γ(0) = x and γ(T ) = y, as considered in [10]. In our case the Lagrangian is defined by L(x, v) = 1 2 |v| 2 + δ t=T /2 Q(x). Moreover, we can compute explicitly the optimal path γ by dividing the transport map m = m 2 • m 1 such that at t = T 2 , and at t = T , Correspondingly, Formally at the minimizer by taking 0 = ∂cT (x,y) ∂z one can recover the equality (4.3), Furthermore, through a straightforward computation, one has Now, letφ * ,Q * be Legendre transforms of convex functionsφ,Q, respectively, i.e.   whereQ * is exactly the Legendre transform ofQ as defined above (4.5). Note that Q * is well defined and C 4 smooth under the assumptions (H0) and (H1).
Note that the terms involving only x or y do not affect the optimal transport map, therefore, we will look at an optimal transport problem with cost c(x, y) =Q * (x + y), (4.6) and seek to maximize the cost functional among all maps s : Ω 0 → Ω T such that s # ρ 0 = ρ T . Note that here we consider the maximization instead of the minimization problem as in [24,15]. When the cost function c is strictly convex as is the case forQ * in (4.6) satisfying (H0)-(H1), it was proved [25,26] that a unique optimal mapping can be determined by the potential functions, that leads to the Monge-Ampère equation with the natural boundary condition (3.5), where p = x + y. Note that the matrix (D 2φ − D 2Q * ) is nonnegative, and D 2Q * is positive definite by condition (H1), which makes Eq. (4.7) elliptic. Note also that in the absence of regularity, one has to understand the solution to (4.7) in the weak "Brenier" sense in Definition 3.1 (see [24]). In our case, ∇Q * (x + y) = ∇φ(x), (4.8) and using the properties of the Legendre transform From the above computations, the initial velocity is given by 2 while ∇φ(x) = z, where z is the mid-point of the trajectory at t = T /2.
We remark that from the uniqueness of v in [9] [in fact, from the duality argument of [9, §3.2] two optimal solutions must have the same density ρ(T /2), thus the uniqueness of v follows since I(ρ, v) is strictly convex in v], a solution of (4.7) is thus the velocity potentialφ. Therefore, we have the following. Theorem 2.1 follows then directly from Proposition 4.1. Additionally, to prove Theorem 2.2 it is equivalent to obtain the regularity of solutions of (4.7) in optimal transportation, that requires appropriate convexity conditions on domains Ω 0 , Ω T , and more importantly some structure conditions on the potential functionQ to be described in the following sections.

Conditions on the Potential Function
From Proposition 4.1, in order to obtain the regularity of the velocity potentialφ, it suffices to consider the optimal transportation with the cost function (4.6). From the results of [24], it is now well understood that the so-called MTW condition (introduced in [15]) is necessary (at least in its weak form) for the regularity of optimal mappings.
First, let us recall the MTW condition in optimal transportation. For a general cost function c(x, y) : R d × R d → R, use the notation In our case, the cost function is given by the potential functionQ * in (4.6). To introduce the analogous MTW condition, denote the matrix D 2Q * (p) = (D 2Q (∇φ)) −1 =: A(z), (5.2) where p = x + y and z = ∇φ, as in (4.5) and (4.2), respectively. Since c(x, y) = Q * (x + y), by differentiation we have From the Legendre transform and (5.2), one has Hence, using Einstein summation one has and thus Therefore, the MTW condition in our case is that for any ξ, η ∈ R d , with ξ⊥η, where c 0 > 0 is a constant. Next, we shall formulate the condition (5.5) in terms of the original potential functionQ. From (5.2), A ij = D 2 ijQ . By differentiating I = AA −1 , we have Optimal transport with discrete long-range mean-field interactions Hence, the left-hand side of (5.5) is D 2 ηη A ξξ = − D 2 ηηQij (Aξ) i (Aξ) j + 2(Aξ) i D ηQir Q rs D ηQsj (Aξ) j .
Comparing with (5.3), one can see that (5.8) is in a similar form in spite of the factor 2.

Regularity of the Potential
It is well known that in order to guarantee some regularity for Eq. (4.7), one needs some notion of convexity of domains. In optimal transport, it has been proved that if the target domain is not c-convex, there exist some smooth densities ρ 0 , ρ T such that the optimal mapping is not even continuous, see [15, §7.3]. For global regularity, one needs both the initial and the target domains to be uniformly c-convex in [19].
In our case, the cost function is c(x, y) =Q * (x + y). Similarly to the c-convexity in optimal transportation, we introduce the following q-convexity for domains. The q-exponential map). Assume the potential functionQ satisfies conditions (H0) and (H1). For x ∈ Ω 0 we define the q-exponential map at x, denoted by I x : R n → R n , such that