Existence of a solution to an equation arising from the theory of Mean Field Games

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ScienceDirect J. Differential Equations 259 (2015) 6573–6643 www.elsevier.com/locate/jde

Existence of a solution to an equation arising from the theory of Mean Field Games ´ ech Wilfrid Gangbo ∗ , Andrzej Swi˛ Georgia Institute of Technology, Atlanta, GA, USA Received 18 May 2015; revised 1 August 2015 Available online 21 August 2015

Abstract We construct a small time strong solution to a nonlocal Hamilton–Jacobi equation (1.1) introduced in [48], the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton–Jacobi equations studied in [2,19,20] and solutions to (1.1). As a consequence we recover the existence of solutions to the First Order Mean Field Games equations (1.2), first proved in [48], and make a more rigorous connection between the master equation (1.1) and the Mean Field Games equations (1.2). © 2015 Elsevier Inc. All rights reserved. MSC: 35R15; 45K05; 49L25; 49N70; 91A13

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Notation and definitions . . . . . . . . . . . . . . . . . . . 2.2. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniqueness of a fix point of Ms [μ] . . . . . . . . . . . . . . . . . 3.1. Elementary properties of Ms [μ] . . . . . . . . . . . . . . 3.2. Differentiability properties of s [μ] on [0, T ] × Td

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* Corresponding author.

´ ech). E-mail addresses: [email protected] (W. Gangbo), [email protected] (A. Swi˛ http://dx.doi.org/10.1016/j.jde.2015.08.001 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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3.3. s-Orbits passing through μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of  in the variables (t, s, q); continuity in μ . . . . . . . . . . . . . . . Minimality properties of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Optimality properties of discrete paths . . . . . . . . . . . . . . . . . . . . . . 5.2. Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Hamilton–Jacobi equation on P(Td ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Semiconvexity/semiconcavity properties of the value function . . . . . . 7. Weak solution to the first order Mean Field equations . . . . . . . . . . . . . . . . 8. Regularity properties of (t, s, q, ·); a discretization approach . . . . . . . . . . 8.1. Spatial derivatives of the discrete master map . . . . . . . . . . . . . . . . . 8.2. Spatial derivatives of the inverse of the master map . . . . . . . . . . . . . 8.3. Regularity properties of the master map . . . . . . . . . . . . . . . . . . . . . 8.4. Regularity properties of the inverse master map . . . . . . . . . . . . . . . . 8.5. First order expansion of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Smoothness properties of the velocity |Vs |2 . . . . . . . . . . . . . . . . . . 9. Strong solutions to the master equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Regularity of u with respect to the μ variable . . . . . . . . . . . . . . . . . 9.2. Existence of a strong solution to the master equation . . . . . . . . . . . . 9.3. Connection with MFG equations and existence of a Nash equilibrium Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 5.

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1. Introduction The theory of Mean Field Games (MFG) analyzes differential games with a large number of players, each player having a very little influence on the overall system. This theory, which encompasses games with a continuum of players, was developed by Lasry–Lions [44–47]. Similar ideas were independently introduced at the same time and studied in the engineering literature by Huang–Caines–Malhamé [36–38,40]. Games with a continuum of players or traders, first appeared in economics, starting with the seminal work of Aumann [5]. Later a theory of non-atomic games was presented in a book by Aumann–Shapley [6]. In this pioneering work, Aumann– Shapley proposed a profound mathematical theory for economics, the potential of which has not yet been fully exploited. The term “Mean Field Games” was introduced by analogy with the mean field models in mathematical physics where the behaviors of many identical particles are analyzed. We refer the readers to [9,12,22,29,32] for several excellent surveys on the theory of MFG and its extensions. In particular, the notes [12] from the lectures of P.-L. Lions [48] have been a great contribution to the field, and have clarified the current state of the theory of MFG. This was the starting point of our study. The theory of MFG has attracted significant attention. In the past five years alone, a large number of manuscripts have been devoted to it, revealing its importance, impact, and possible applications (see e.g. [7,8,15,23–28,30,31,33–35,39,41–43,49–52]). In light of the publications [44–47], we restrict our study to the simplest framework of games: those with identical players. Our effort will be devoted mainly to the study of the master equation of MFG (1.1); only a small part of the manuscript deals with the MFG equations (1.2) which were studied in [46,48,12]. Our main result establishes the short time existence of a regular solution to (1.1).

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Let us denote by P(Td ) the set of probability measures on the d-dimensional torus Td , let T > 0 be a real number, and let F, u∗ : Td × P(Td ) → R be Lipschitz functions. The objective is to find a continuous function u : [0, T ] × Td × P(Td ) → R such that 

  ∂s u(s, q, μ) + ∇μ u(s, q, μ), ∇q u(s, ·, μ) μ + u(0, ·, ·) = u∗ (·, ·)

|∇q u(s,q,μ)|2 2

+ F (q, μ) = 0,

(1.1)

is satisfied in some sense. Here, ∇μ u stands for the Wasserstein gradient of u and we have set  ∇μ u(s, q, μ), ∇q u(s, ·, μ)μ =

∇μ u(s, q, μ)(z) · ∇q u(s, z, μ)μ(dz). Td

We will call (1.1) the master equation of the theory of MFG. A heuristic derivation of (1.1) as the limit of a large system of Hamilton–Jacobi equations arising from Nash equilibria in feedback form for many players, can be found in [48] (see also [12]). Furthermore, [48] describes the connection between (1.1) and the first order MFG equations ⎧ 2 ⎨ ∂t U (t, q) + |∇U (t,q)| + F (q, σt ) = 0 2

∂ σ + ∇ · (σ ∇U (t, q)) = 0 in D (0, T )) × Td t ⎩ t t U (0, q) = u∗ (q, σ0 ), σT = μ.

(1.2)

In (1.2), the first equation is supposed to be satisfied in the viscosity sense and U represents the value function of a typical player. The second equation is supposed to be satisfied in the distribution sense and σt , represents the probability distribution of all the players at time t . The measures μ and σt in (1.2) are also supposed to be absolutely continuous with respect to the Lebesgue measure for every t . The main difficulty in dealing with (1.1) is the following. Observe that for each (s, q, μ) fixed, the knowledge of ∂s u, ∇μ u and ∇q u at (s, q, μ) is not sufficient to verify that the equation is satisfied since we need the knowledge of ∇q u(s, z, μ) for all z ∈ Td to fully describe (1.1). In other words, (1.1) is non-local in ∇q u. This difficulty is coupled with the infinite dimensional character of the equation. Interpreting in what weak sense (1.1) may be satisfied has remained a puzzle so far. We try to unravel it by providing a possible definition in the current manuscript (see Definition 7.3). More importantly, we prove the existence of a strong solution to (1.1) for a short time, assuming that the data are sufficiently smooth. We hope this work will help uncover some groundbreaking facts and improve our understanding of the theory of MFG. Not to overshadow the main ideas with technical details, we have opted in this manuscript to restrict the study of (1.1) to a particular – nevertheless important – class of F ’s. More precisely, we choose φ ∈ C 3 (Td ) and consider F (q, μ) = φ ∗ μ(q).

(1.3)

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However we stress that the approach developed in this paper can be carried out for a wider and quite general class of functionals. The starting point of our work is the value function U which is the unique metric viscosity solution, in the sense of [19] and [20] (see also [2] and [18]), to the Hamilton–Jacobi equation 

 ∂t U + 12 ||∇μ U||2μ + Td 12 φ ∗ μdμ = 0 U (0, ·) = U∗ on P(Td ).

in (0, T ) × P(Td )

(1.4)

We draw the attention of the reader to the fact that having the coefficient 1/2 in front of φ in (1.4) and not having it in (1.3) is not a typo. According to the well-established theory of endowing the set of probability measures P(Td ) with a weak Riemannian structure (see e.g. [4]), the Wasserstein gradient ∇μ U of U at μ ∈ P(Td ) is an element of Tμ P(Td ) := ∇C ∞ (Td )

L2 (μ)

,

(1.5)

the tangent space to P(Td ) at μ. Hence ∇μ U(t, μ) : Td → Td is a map which, formally at least, is the gradient of a function u(t, ·, μ): ∇q u(t, q, μ) = ∇μ U (t, μ)(q).

(1.6)

One of the tasks of the current manuscript will include finding a function u satisfying (1.6) which will also satisfy (1.1). The identity (1.6) linking (1.1) to (1.4), appears to be an unexpected connection between two different directions of research which, over the past several years, have been pursued by different research groups using different methods. Indeed, so far the study of (1.4) was primarily motivated by aspects of fluids mechanics (see e.g. [17–20]). The lectures of P.-L. Lions [48] presented in the notes by Cardialaguet [12] seem to be the first to imply a connection between these two directions. We stress here that the readers should not be misled to think that they need a prior knowledge of the various viscosity solution concepts introduced in [2,18–20] to grasp the content of this manuscript. We have mentioned the works on metric viscosity solutions just to emphasize that there is a connection between (1.1) and (1.4) via the identity (1.6), which could be explored in future studies. The cornerstone of our work, besides establishing identity (1.6), is a good understanding of the regularity properties with respect to the μ variable, of the inverse Xst [μ] of the map st [μ]. The latter map is defined uniquely for small enough T and s ∈ [0, T ] by the system of differential equations ⎧ t

t t d ⎪ ⎨ ∂tt s [μ](q) = −∇q F s [μ](q), s [μ]# μ , on (0, T ) × T s d s [μ](q) = q on T ⎪ ⎩ ∂  0 [μ](q) = ∇ u  0 [μ](q),  0 [μ] μ)

on Td . t s q ∗ # s s

(1.7)

We will often write (t, s, q, μ) for st [μ](q). The regularity property of  in the variables (t, s, q) and the invertibility property of (t, s, ·, μ) are obtained by standard methods. However,

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the regularity property with respect to μ of the inverse of (t, s, ·, μ) is subtle. We overcome this obstacle by first discretizing st [μ](q) in its μ-variable and then studying the maps (t, q, x) ∈ (0, T ) × Td × (Td )n → (t, st [μx ](q), x), where we have set 1 δxi , n n

μx :=

x = (x1 , . . . , xn ).

i=1

The determinant of the Jacobian of ∇t,q,x st [μx ](q) is shown to be controlled in terms of the finite dimensional determinant det ∇q st [μx ](q). This allows us to apply the Inverse Function Theorem and then obtain bounds on partial derivatives of the inverse of this map using the bounds on the partial derivatives of S(t, s, q, x) := st [μx ](q). This task is completed in Section 8. In Sections 5 and 6 we show that, if μ ∈ P(Td ) and if s > 0 is small enough, the infimum in the variational problem (6.2) related to the Hamilton–Jacobi equation (1.1) is attained by a path (σ, v), where σt = st [μ]# μ,

vt = ∇μ U (t, σt ).

In Section 7 we construct a function u(t, q, μ) such that the pair U (t, q) = u(t, q, σt ) and σ satisfy the First Order Mean Field Games equations (1.2). We also show that u is a solution to (1.1) in some weak sense (see Lemma 7.1 and Definition 7.3). The statement (1.6) is one of the things we prove at this stage of the analysis. Then, in Section 9 we prove regularity properties of the function u for small times t that allow us to differentiate u with respect to each variable and show that u satisfies (1.1) pointwise. We call such a function a strong solution of (1.1). Uniqueness of strong solutions remains open. Finally, in Subsection 9.3 we make a rigorous link between strong solutions to the master equation (1.1) and the Mean Field Games equations (1.2) by showing in Lemma 9.9 that any strong solution u to (1.1) allows to construct a pair (U, σ ) which solves (1.2), and argue that u also allows to construct an analogue of a Nash equilibrium for a game with a continuum of players. After the manuscript was completed we learned about the papers [10,13] which deal with formal derivation of the Master Equations in both deterministic and stochastic cases and their analysis. Also during the second submission of the paper a referee pointed out preprints [11,14] which deal with classical solutions of Master Equations for stochastic Mean Field Games. 2. Preliminaries 2.1. Notation and definitions Throughout this manuscript, Td = Rd /Zd is the d-dimensional torus. When there is no possible confusion we identify an element of the quotient space Td with the unique q ∈ [0, 1)d . We denote by |q ∗ − q|Td the distance on Td between q ∗ , q ∈ Td . The Euclidean distance between

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q ∗ , q ∈ Rd is denoted by |q ∗ − q|. If ξ ∈ Rd×m we denote |ξ |2 =

m d

ξij2 .

i=1 j =1

We denote by Id : Td → Td the identity map and by Id the d × d identity matrix. Definition 2.1. Let f : Rd → R. (i) By f : Td → Rk we mean that f : Rd → R and if q, q ∗ ∈ Rd are such that q − q ∗ ∈ Zd , then f (q ∗ ) = f (q). (ii) By f : Td → Td we mean that if q, q ∗ ∈ Rd are such that q − q ∗ ∈ Zd then f (q ∗ ) − f (q) ∈ Zd . (iii) By X ∈ C(Td ; Td ) we mean that X : Rd → Rd is continuous and X : Td → Td .

If T > 0 and S ∈ W 2,∞ (0, T ) × Td ; Td , unless explicitly stated otherwise, ∇tq S := ∂t ∇q S, ∇qt S := ∇q ∂t S, etc., denote the distributional derivatives of S. Since for instance, the distributional derivatives ∂t ∇q S and ∇q ∂t S coincide, we denote them by ∇tq S = ∇qt S. Since the distributional derivatives coincide almost everywhere with the pointwise derivatives, expressions such as ||∇q S||∞ will be used to denote the essential supremum of the function |∇q S|. Given two metric spaces S1 , S2 and a map S : [0, T ] × [0, T ] × S1 → S2 we use the notation S(t, s, ξ ) = Sst (ξ ),

(t, s, ξ ) ∈ [0, T ] × [0, T ] × S1 .

If S : [0, T ] × Td × P(Td ) → Rd is a bounded Borel function, the smallest number A such that |S(t, q, μ)| ≤ A for almost every (t, q) ∈ [0, T ] × Td and all μ ∈ P(Td ) is denoted by ||S||∞ . We denote by P2 (Rd ) the set of Borel probability measures on Rd with finite second moments. On P2 (Rd ) we can define a class of equivalence (cf. e.g. [21]): we say that μ, ν ∈ P2 (Rd ) are equivalent if for all f ∈ C(Td ) we have 

 f (q)μ(dq) =

Rd

f (q)ν(dq). Rd

We use the notation 

 f (q)μ(dq) :=

Td

f (q)μ(dq). Rd

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The quotient of P2 (Rd ) by the equivalence relation is P(Td ), the set of Borel probability measures on Td . The set P(Td ) has been amply studied in [21], as the quotient space of P2 (Rd ), and so, we refer to that manuscript for more details. We just recall that any measure μ ∈ P2 (Rd ) yields a measure μ¯ on [0, 1)d which is defined by μ(B) ¯ =

μ(B + k)

k∈Zd

for a Borel B ⊂ [0, 1)d . Given μ ∈ P(Td ), we denote by L2 (μ) the set of Borel maps ξ : Td → Rd which are square integrable and we set  ||ξ ||2μ = |ξ |2 μ(dq). Td

Given a Borel map X : Td → Td and μ ∈ P(Td ), we denote by X# μ the push forward of μ by X.

Definition 2.2. (Cf. [4].) Let σ ∈ AC2 0, T ; P(Td ) . We say that a Borel vector field v : (0, T ) × Td → Rd is a velocity for σ if t → ||vt ||σt is in L2 (0, T ) and ∂t σ + ∇ · (σ v) = 0 in the sense of distributions on (0, T ) × Td . The latter statement means that for every f ∈ Cc1 (0, T ) × Td T  0



∂t f (t, q) + vt (q)∇f (t, q) σt (dq) dt = 0.



Td

When x1 , · · · , xn ∈ Td we set x = (x1 , · · · , xn ) and 1 δxi . n n

μx =

i=1

Definition 2.3. Given μ, ν ∈ P(Td ), we define (μ, ν) to be the set of measures γ ∈ P(Td × Td ) which have μ as the first marginal, and ν as the second marginal. We denote by 0 (μ, ν) the set of γ ∈ (μ, ν) such that   W22 (μ, ν) := min |r − q|2Td γ¯ (dq, dr) = |r − q|2Td γ (dq, dr). γ¯ ∈ (μ,ν) Td ×Td

Td ×Td

Recall that P(Td ) endowed with the Wasserstein distance W2 is a compact metric space and a sequence {μk }k ⊂ P(Td ) converges to μ in the Wasserstein metric if and only if its converges narrowly.

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Definition 2.4. If μ ∈ P(Td ), we define Tμ P(Td ) to be the closure in L2 (μ) of the set ∇C ∞ (Td ) := {∇f : f ∈ C ∞ (Td )}. Let F : P(Td ) → R. We define the Lagrangian L and the Hamiltonian H by 1 L(μ, ξ ) := ||ξ ||2μ − F(μ), 2

1 H(μ, ξ ) := ||ξ ||2μ + F(μ) 2

(2.1)

for μ ∈ P(Td ) and ξ ∈ L2 (μ). The assumptions on F will be given in Subsection 2.2. Recall that a function ψ : Rm → R is λ-convex (respectively, λ-concave) if ψ(x) − λ/2|x|2 is convex (respectively, concave). Such functions are called semiconvex (respectively, semiconcave). By analogy, the concept of λ-convex functions on P(Td ) was introduced in [4]. We refer the reader to the same book for more on the Wasserstein space, absolutely continuous curves in metric spaces, etc. Following [18] we give a definition of the sub-differential which in general does not coincide with that of [4] except for λ-convex functions. Definition 2.5. Let G : P(Td ) → R and let μ ∈ P(Td ). (i) We say that ξ belongs to the subdifferential of G at μ and we write ξ ∈ ∂· G(μ) if ξ ∈ L2 (μ) and  G(ν) − G(μ) ≥

sup

γ ∈ o (μ,ν) Td ×Td



ξ(q) · (r − q)γ (dq, dr) + o W2 (μ, ν)

∀ν ∈ P(Td ). (2.2)

(ii) We say that ξ belongs to the superdifferential of G at μ and we write ξ ∈ ∂ · G(μ) if −ξ ∈ ∂· (−G)(μ). The unique element of minimal norm in ∂ · G(μ) belongs to Tμ P(Td ) and is called the gradient of G at μ and is denoted by ∇μ G(μ). (iii) We say that G is differentiable at μ if both ∂· G(μ) and ∂ · G(μ) are non-empty. In that case (see e.g. [18]) both sets coincide and ∂· G(μ) ∩ Tμ P(Td ) = ∂ · G(μ) ∩ Tμ P(Td ) = {∇μ G(μ)}. Remark 2.6. Here are few remarks. (i) We refer the reader to Remark 3.2 of [18] for property (iii) in Definition 2.5. (ii) Thanks to Proposition 8.5.4 of [4], note that (2.2) holds for ξ if and only if it holds for any ξ0 ∈ L2 (μ) such that ξ0 − ξ belongs to the orthogonal complement of Tμ P(Td ) in L2 (μ). Rephrasing, if (2.2) holds for ξ0 ∈ L2 (μ) then it holds for ξ defined as the orthogonal projection of ξ0 onto Tμ P(Td ). Remark 2.7 (Basic properties of the determinant). Let ξ = (ξij ) ∈ Rd×d and denote by ξ¯ = (ξ¯ij ) the matrix of its cofactors.

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(i) We can write ξ = LQ where L is lower triangular and Q is orthogonal. Thus, ξ ξ T = LLT and so, 2

2

| det ξ | d = | det L| d ≤

2 + · · · + l2 l11 |L|2 |ξ |2 dd ≤ = . d d d

(ii) We have ∂ξij det ξ = ξ¯ij and so, by (i), if d > 1 then |∂ξij det ξ | ≤ √

|ξ |d−1 d−1

d −1

(2.3)

and, using the fact that d 2 ≤ 4(d − 1)d−1 we conclude that |∇ξ det ξ | ≤ 2|ξ |d−1 .

(2.4)

If d = 1 then det ξ = ξ . In that case (2.4) continues to hold. 2.2. Assumptions We state here general assumptions that will be used in the manuscript. In the second part of the paper (from Section 6 on) we will further assume that the functions F, F, u∗ , U∗ have particular forms. Let κ ≥1

(2.5)

be a given constant. We assume we have a differentiable function F : Td × P(Td ) → R and a differentiable κ-Lipschitz function F : P(Td ) → R such that for any q ∈ Td and any μ ∈ P(Td ), ∇q F (q, μ) = ∇μ F(μ)(q),

(2.6)

and           ∇μ F(μ)(q) · (y − q)γ (dq, dy) ≤ κ |q − y|2Td γ (dq, dy), (2.7) F(ν) − F(μ) −     Td ×Td Td ×Td for all ν ∈ P(Td ) and all γ ∈ (μ, ν) We further assume that

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∇q F (q, μ), ∇qq F (q, μ), ∇qqq F (q, μ)

exist and are continuous,

∇q F ∞ , ∇qq F ∞ , ∇qqq F (q, μ)∞ ≤ κ,

(2.8) (2.9)

and ∇q F

is κ-Lipschitz.

(2.10)

We assume to be given a κ-Lipschitz function u∗ : Td × P(Td ) → R such that |u∗ | ≤ κ.

(2.11)

We assume there is a differentiable κ-Lipschitz function U∗ : P(Td ) → R such that for any q ∈ Td and any μ ∈ P(Td ), ∇q u∗ (q, μ) = ∇μ U∗ (μ)(q),

(2.12)

         ∇μ U∗ (μ)(q) · (y − q)γ (dq, dy) U∗ (ν) − U∗ (μ) −     Td ×Td  |q − y|2Td γ (dq, dy), ≤κ

(2.13)

and

Td ×Td

for all ν ∈ P(Td ) and all γ ∈ (μ, ν). We further assume that ∇q u∗ , ∇qq u∗ , ∇qqq u∗

exist and are continuous,

||∇q u∗ ||∞ , ||∇qq u∗ ||∞ , ||∇qqq u∗ ||∞ ≤ κ,

(2.14) (2.15)

and ∇q u∗

is κ-Lipschitz.

(2.16)

Remark 2.8. Observe that the requirement on F in (2.7) is more restrictive than 2κ-geodesic convexity and 2κ-geodesic concavity (see Proposition 4.2 of [3]) since we do not require that γ ∈ 0 (μ, ν). A similar remark applies to (2.13).

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If s ∈ [0, T ] and σ ∈ AC2 0, T ; P(Td ) has v as a velocity, we define the augmented action s A(s; σ, v) :=

L(σl , vl )dl + U∗ (σ0 ). 0

For s ∈ [0, T ] and μ ∈ P(Td ), we define the map



Ms [μ] : C [0, T ] × Td ; Td → C [0, T ] × Td ; Td by

Ms [μ](S)(t, q) = q + (t − s)∇q u∗ S 0 (q), S#0 μ s +

l dl

t



∇q F S τ (q), S#τ μ dτ ,

(2.17)

0

where we used the notation S τ for S(τ, ·). Example 2.9. Let φ, U 0 , U 1 ∈ C 3 (Td ) be such that φ and U 1 are even and (6.1) holds. For any q ∈ Td , μ ∈ P(Td ) we set u∗ (q, μ) = U 0 (q) + U 1 ∗ μ(q), F (q, μ) = φ ∗ μ(q),  0 1 1

U∗ (μ) = U + U ∗ μ (y)μ(dy), 2 Td

1 F(μ) = 2

(2.18)

 φ ∗ μ(y)μ(dy).

(2.19)

Td

We have ∇q u∗ (q, μ) = ∇U 0 (q) + ∇U 1 ∗ μ(q),

∇q F (q, μ) = ∇φ ∗ μ(q),

and it can be shown, using techniques of [4], that F , F , u∗ and U∗ satisfy all the assumptions of this section. 3. Uniqueness of a fix point of Ms [μ] Throughout this section, T > 0 is a prescribed number. Further restrictions on T will be placed later. We denote CT := T (1 + T ).

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3.1. Elementary properties of Ms [μ] Let

S ∈ W 1,∞ [0, T ] × Td ; Td . Using the notation S t = S(t, ·) we have



∂t Ms [μ](S) (t, q) = ∇q u∗ S 0 (q), S#0 μ −

t ∇q F (S l (q), S#l μ)dl,

(3.1)

0



∂tt M[μ](S) (t, q) = −∇q F (S t (q), S#t μ)

(3.2)

and

∇q Ms [μ](S)(t, q) = Id + (t − s)∇qq u∗ S 0 (q), S#0 μ ∇q S 0 (q) s +

l ∇qq F (S τ (q), S#τ μ)∇q S τ (q)dτ.

dl t

(3.3)

0

Hence

∇tq Ms [μ](S)(t, q) = ∇qq u∗ S 0 (q), S#0 μ ∇q S 0 (q) t −



∇qq F S τ (q), S#τ μ ∇q S τ (q)dτ.

(3.4)

0



Lemma 3.1. Let S ∈ W 2,∞ [0, T ] × Td ; Td and let A > 0 be such that ||∇q S||∞ , ||∇qq S||∞ ≤ A. Then for any μ ∈ P(Td ) and any s ∈ [0, T ] we have: (i) √ ||Ms [μ](S)||∞ ≤

d + κCT , ||∂t Ms [μ](S)||∞ ≤ κ(1 + T ), ||∂tt Ms [μ](S)||∞ ≤ κ. (3.5) 2

(ii) ||∇q Ms [μ](S)||∞ ≤

√

d + κACT , ||∇tq Ms [μ](S)||∞ ≤ κA(1 + T ).

(iii) ||∇qq Ms [μ](S)||∞ ≤ κA(1 + A)CT .

(3.6)

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Proof. To show (i) we use (2.9) and (2.15) to obtain the first inequality in (i). We use the formulas for first and second derivatives of Ms [μ](S) with respect to t , given by (3.1) and (3.2) and then use (2.9) and (2.15) to obtain the second and third inequalities in (i). Similarly, the inequalities in (3.6) are obtained from (3.3) and (3.4), using (2.9) and (2.15). To get the inequality in (iii) we differentiate (3.3) with respect to q and again use (2.9) and (2.15) and the assumptions on S. 2 We also suppose that A > 0 and T > 0 are such that √ 3κ d ≤ A,

2T , 3κCT < 1,

√ 4κT A( d + 1)d−1 ≤ 1.

(3.7)

We observe that the second and third inequalities above give κ(1 + A)CT ≤ 1.

(3.8)

Lemma 3.2. Let μ ∈ P(Td ) and let

 ∈ W 1,∞ (0, T ) × (0, T ) × Td ; Td ,



¯ s, ·) := Ms [μ] (s, ·, ·) . (·,

Then   √  T ≤ κ 1 + T + 2T 1 +  ∂ s ∞ ∞ 2

  ∂s  ¯ and

  ∂s ∂t  ¯

∞

≤

√ 2κ(1 + T )∂s ∞ .

Proof. Since Z → Z# μ is a 1-Lipschitz map of C(Td ; Td ) into P(Td ) and ∇q u∗ , ∇q F are κ-Lipschitz, we conclude that maps

Z → ∇q u∗ Z(q), Z# μ and



Z → ∇q F Z(q), Z# μ

√ are 2κ-Lipschitz for q ∈ Td fixed. We use this to obtain the first inequality. The second inequal¯ ity is obtained in a similar manner applying the above arguments to the formula for ∂t . 2 Remark 3.3. The following hold: (i) The map Z → Z# μ is 1-Lipschitz of C(Td ; Td ) into P(Td ). (ii) If Z : Td → Td is l-Lipschitz, so is μ → Z# μ. As a consequence if Z ∈ C(Td ; Td ) then ζZ : μ → Z# μ is a continuous map of P(Td ) into itself P(Td ). Therefore, if S ∈ C [0, T ] ×

Td ; Td , then ζS(0,·) is a continuous map of P(Td ) into itself. Proof. (i) If S1 , S2 ∈ C(Td ; Td ) then

γ := (S1 × S2 )# μ ∈ S1 # μ, S2 # μ ,

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and so,

W22 S1 # μ, S2 # μ ≤ S1 − S2 2μ ≤ S1 − S2 2∞ . (ii) Let Z : Td → Td be l-Lipschitz and let μ1 , μ2 ∈ P(Td ). If γ ∈ 0 (μ1 , μ2 ) then γ¯ := (Z × Z)# γ ∈ (Z# μ1 , Z# μ2 ). Hence,

W22 Z# μ1 , Z# μ2 ≤

 |q − r|2Td γ¯ (dq, dr)

Td ×Td



=

|Z(x) − Z(y)|2Td γ (dx, dy)

Td ×Td

≤ l 2 W22 (μ1 , μ2 ).

(3.9)

This proves that ζZ is l-Lipschitz. Let now Z ∈ C(Td ; Td ) and let {Z k }k be a sequence of Lipschitz functions that converges uniformly to Z on Td . By (i) W2 (ζZ μ, ζZk μ) ≤ Z − Z k ∞ and so, ζZ is continuous as a uniform limit of Lipschitz maps. 2 Remark 3.4. The following hold: (i) By assumption ∇q F and ∇q u∗ are κ-Lipschitz. Since, by Remark 3.3(i), Z → Z# μ is a 1-Lipschitz map of C(Td ; Td ) into P(Td ), we conclude that if s ∈ (0, T ] then Ms [μ] : C [0, T ] × Td ; Td → C [0, T ] × Td ; Td is Lipschitz continuous with the Lipschitz con√ stant which or equal to 2κCT < 1. Thus Ms [μ] is a contraction.

is less than (ii) If S ∈ C [0, T ] × Td ; Td then, by Remark 3.3(ii), μ → S#0 μ is a continuous map of

P(Td ) into itself. Since ∇q u∗ is κ-Lipschitz, we obtain that μ → ∇q u∗ S 0 (q), S#0 μ is to conclude that the map continuous. We use (i) and the fact that ∇q F is κ-Lipschitz (s, S, μ) → Ms [μ](S) is a continuous map of [0, T ] × C [0, T ] × Td ; Td × P(Td ) into

C [0, T ] × Td ; Td . Definition 3.5. We define

(i) CA to be the set of S ∈ C [0, T ] × Td ; Td such that √ S∞ ≤

d +1 , ∂t S∞ ≤ 2κ, 2

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and 3 ∂tt S∞ ≤ κ, ∇q S∞ , ∇qq S∞ ≤ A, ∇tq S∞ ≤ κA. 2

(ii) We define CA∗ to be the set of  ∈ C [0, T ] × [0, T ] × Td ; Td such that for every s ∈ [0, T ], (·, s, ·) ∈ CA and ∂s ∞ ≤ A,

∂ts ∞ ≤

√

2κ(1 + T )A.

Lemma 3.6. The following hold:

(i) CA is a compact set in C [0, T ] × Td ; Td . (ii) If s ∈ [0, T ] and μ ∈ P(Td ) then Ms [μ] maps CA into itself. Proof. (i) We omit the proof of (i) because it is elementary. (ii) Let S ∈ CA . Since 3κCT ≤ 1, we use Lemma 3.1(i) and the fact that T ≤ 1 to obtain √ Ms [μ](S)∞ ≤

d +1 , ∂t Ms [μ](S)∞ ≤ 2κ, ∂tt Ms [μ](S)∞ ≤ κ. 2

(3.10)

√ By (3.7) since κ ≥ 1 we have 3 d ≤ A. We use the latter inequality in Lemma 3.1(ii) and use the fact that 3κCT ≤ 1 to obtain ∇q Ms [μ](S)∞ ≤ A.

(3.11)

The inequality κ(1 + A)CT ≤ 1 implies κA(1 + A)CT ≤ A. This, together with Lemma 3.1(iii), gives ∇qq Ms [μ](S)∞ ≤ A.

(3.12)

Since T ≤ 1, we obtain 3 ∇qt Ms [μ](S)∞ ≤ κA(1 + T ) ≤ κA. 2 This, together with (3.10), (3.11) and (3.12), yields Ms [μ](S) ∈ CA .

2

Lemma 3.7. Let μ ∈ P(Td ) and let  ∈ CA∗ . Define

¯ s, q) = Ms [μ] (·, s, ·) (t, q) (t, ¯ ∈ C∗ . Then  A

∀(t, s, q) ∈ [0, T ] × [0, T ] × Td .

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¯ s, ·) ∈ CA . By Proof. Since for any s ∈ [0, T ] we have (·, s, ·) ∈ CA , Lemma 3.6 yields (·, Lemma 3.2, since  ∈ CA∗ and T < 1/2, we have √  3 2 ≤ κ + T A . ∞ 2 2

  ∂s  ¯

3

The first inequality in (3.7) ensures that A ≥ 3 and then the third inequality there gives 12κT ≤ 1. Therefore, using again the first inequality in (3.7), √  A 3√2 1 3 2 κ + TA ≤ + · A < A. 2 2 2 2 12   ¯  ≤ A to complete the We use the second inequality in Lemma 3.2 and the fact that ∂s  ∞ proof. 2 3

d Theorem 3.8. Let μ ∈

P(T ) and let 0 ≤ s ≤ T . Then Ms [μ] admits a unique fixed point ds [μ]

d d in C [0, T ] × T ; T . Furthermore, s [μ] belongs to every closed subset of C [0, T ] × T ; Td which is invariant under Ms [μ]. As a consequence we have:

(i) For any k ∈ Zd , s [μ](t, q + k) = s [μ](t, q) + k. (ii) s [μ] ∈ CA .

Proof. Since Ms [μ] is a contraction in C [0, T ] × Td ; Td , it has a unique fixed point s [μ].

Any closed subset C of C [0, T ] × Td ; Td is also a complete metric space. Hence, if C is invariant under Ms [μ], there must be a unique fixed point of Ms [μ] in C which, by uniqueness, must be equal to s [μ]. Since, by Lemma 3.6, CA is a compact set invariant under Ms [μ], we thus obtain (ii).

(i) If k ∈ Zd , the set C which consists of S ∈ C [0, T ] × Td ; Td such that S t (q + k) = S t (q) for all t ∈ [0, T ] and all q ∈ Td , is closed. To show that s [μ] ∈ C, its remains to show that C is invariant under Ms [μ]. If S ∈ C, using the facts that ∇q u∗ (·, ν), ∇q F (·, S#τ μ) : Td → Rd , we obtain Ms [μ](S) ∈ C.

2

Lemma 3.9. Let μ ∈ P(Td ), and let  0 ∈ CA∗ . Define inductively sk = Ms [μ](sk−1 ) for k ≥ 1. Then the sequence { k } converges uniformly to [μ], where [μ](t, s, q) = s [μ](t, q), and [μ] ∈ CA∗ . for each s ∈ Proof. By Lemma 3.7, an induction argument shows that  k ∈ CA∗ . In particular [0, T ],  k (·, s, ·) ∈ CA . Recall that, by Lemma 3.6, CA is a compact set in C [0, T ] × Td ; Td and Ms [μ] maps CA into CA . Since Ms [μ] is a contraction we conclude that { k (·, s, ·)}k converges uniformly on [0, T ] × Td to s [μ]. We use the equicontinuity of { k }k to infer its uniform

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convergence on [0, T ] × [0, T ] × Td to [μ]. Since CA∗ is closed for the uniform convergence, [μ] ∈ CA∗ . 2 Definition 3.10. Under the assumptions of Theorem 3.8, we define s [μ] to be the unique fixed point of Ms [μ] and write st [μ] in place of [μ](t, s, ·). We will sometimes also use the notation (t, s, q, μ). Lemma 3.9 ensures that we can always assume [μ] ∈ CA∗ . 3.2. Differentiability properties of s [μ] on [0, T ] × Td In the sequel, we assume that T > 0, A > 0 and (3.7) holds. Remark 3.11. Let ξ ∈ Rd×d be such that |ξ | ≤ 3/2κA. For any s ∈ [0, T ], we have: (i) 3 | det(I + sξ ) − 1| ≤ . 4 (ii) I + τ sξ is invertible and   √   (I + τ sξ )−1  ≤ cd := 8( d + 1)d−1 . Proof. (i) We use (3.7) to obtain that 3/2κT A < 1 and so, for any τ ∈ [0, 1], √ 3 |I + τ sξ | ≤ |I | + κT A ≤ d + 1. 2

(3.13)

We apply the mean value theorem to det(I + sξ ), to obtain τ ∈ [0, 1] such that | det(I + sξ ) − det I | = s|∇ξ det(I + τ sξ ) · ξ |. We then apply (2.4) and (3.7) to conclude that √ 3 | det(I + sξ ) − det I | ≤ T |ξ |2|I + τ sξ |d−1 ≤ 3κT A( d + 1)d−1 ≤ . 4 (ii) By (i), I + sξ is invertible. Since

−1

(I + sξ )

T ∇ξ det(I + sξ ) = , det(I + sξ )

we use (i), (2.4) and (3.13) to conclude that √   (I + sξ )−1  ≤ 8|I + sξ |d−1 ≤ 8( d + 1)d−1 .

2

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Remark 3.12. If s ∈ [0, T ] and μ ∈ P(Td ) then s [μ] is the unique solution to the system of differential equations   ⎧ t [μ](q) = −∇ F  t [μ](q),  t [μ] μ ⎪  on (0, T ) × Td ∂ tt q # ⎪ s s s ⎨ ss [μ](q) = q on Td   ⎪ ⎪ ⎩ ∂  0 [μ](q) = ∇ u  0 [μ](q),  0 [μ] μ on Td . t s q ∗ # s s

(3.14)

Lemma 3.13. Let μ ∈ P(Td ). (i) For s ∈ [0, T ] and t ∈ [0, T ] we have 4 det ∇q st [μ] ≥ 1. (ii) For s ∈ [0, T ] and t ∈ [0, T ], st [μ] : Td → Td is a diffeomorphism whose inverse is denoted by Xst [μ]. (iii) There exists a constant CA independent of s and μ such that ∂t s [μ]W 2,∞ (0,T )×Td , Xs [μ]W 2,∞ (0,T )×Td , ∂s Xs [μ]∞ , ≤ CA Proof. Fix t ∈ [0, T ]. (i) Use the second equation in (3.14) to write st [μ] = Id + T ζ

and so

∇q st [μ] = Id + T ξ,

where, t Tζ =

t ∂t sτ [μ]dτ

and T ξ =

s

∇tq sτ [μ](q)dτ. s

By Theorem 3.8, s [μ] ∈ CA and so, ξ ∞ ≤ 3κA/2. We apply Remark 3.11 to obtain (i) and (∇q st [μ])−1 ∞ ≤ cd .

(3.15)



s [μ] ∈ CA ⊂ W 2,∞ (0, T ) × Td ; Td

(3.16)

(ii) We use the fact that

and the Sobolev Embedding Theorem to conclude that

s [μ] ∈ C 1 (0, T ) × Td ; Td . Since s [μ] ∈ CA implies that T ζ (t, ·) is T A-Lipschitz, and the last inequality in (3.7) yields T A < 1, we conclude that Id + T ζ is one-to-one. Let R > 1 and let y ∈ BR−1 (0), the ball of radius R − 1 centered at the origin. We use s [μ] ∈ CA to obtain T ζ ∞ < 1 and so, for all q on the boundary of the bigger ball BR (0), ss [μ](q) = y for any s ∈ [0, T ]. Therefore,   f (l) := deg sl [μ], BR (0), y

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the topological degree of sl [μ] is well defined at y ∈ BR−1 (0) (see e.g. [16]). Since f is a continuous function which assumes only integer values, we conclude that f (l) = f (0) = 1. This proves that the range of sl [μ] contains BR−1 (0). Since R > 1 is arbitrary, we conclude that the range of sl [μ] contains Rd . In particular, taking into account that we have already proved that st [μ] is one-to-one, when l = t we obtain that st [μ] : Rd → Rd is a bijection of class C 1 . This, together with 4 det ∇q st [μ] ≥ 1 (by (i)), implies that Xst [μ] : Rd → Rd , the inverse of st [μ], is of class C 1 and satisfies   −1 adj ∇q st [μ]   ◦ Xst [μ]. ∇q Xst [μ] = ∇q st [μ] ◦ Xst [μ] = t det ∇q s [μ] 

(3.17)

Here, if E is a square matrix, adj(E) is the transposed matrix of the cofactors of E. (iii) By (3.15) and (3.17) ∇q Xs [μ]∞ ≤ cd . Direct computations reveal that ∂t Xst [μ] = −∇q Xst [μ] ∂t st [μ] ◦ Xst [μ],

∂s Xst [μ] = −∇q Xst [μ] ∂s st [μ] ◦ Xst [μ].

(3.18)

Thus, using the inequality ∇q Xs [μ]∞ ≤ cd and the fact that [μ] ∈ CA∗ , we obtain the third inequality in (iii). Recall that since s [μ] is a fixed point for Ms [μ] we have

∂t st [μ](q) = ∇q u∗



s0 [μ]q, (s0 [μ])# μ



t −

  ∇q F sτ [μ]q, sτ [μ]# μ dτ.

(3.19)

0

We can now differentiate both sides of (3.19) with respect to t, q and use (2.9), (2.15), Remark 3.3 and the fact that s [μ] ∈ CA to obtain a constant CA , independent of s and μ, such that the first inequality in (iii) holds. Finally we use again s [μ] ∈ CA and differentiate the expressions in (3.17) and (3.18) with respect to t, q, to obtain that the second derivatives of Xs [μ] are bounded by a constant CA independent of μ or s. 2 3.3. s-Orbits passing through μ As in Subsections 3.1 and 3.2, we assume that T > 0, A > 0 are such that (3.7) holds. Given s ∈ [0, T ] we define the s-Orbits through μ by Os [μ] = {st [μ]# μ | t ∈ [0, T ]}. Definition 3.14. For t ∈ [0, T ], s ∈ [0, T ], q ∈ Td and μ ∈ P(Td ) we define Vst [μ] := ∂t st [μ] ◦ Xst [μ].

(3.20)

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Lemma 3.15. Increasing the value of CA we obtain that for all μ ∈ P(Td ) sup Vs [μ]W 2,∞ (0,T )×Td ,

s∈[0,T ]

∂s V[μ]∞ ,

≤ CA .

Proof. Since Vst [μ] := ∂t st [μ] ◦ Xst [μ], the first inequality follows directly from Lemma 3.13. We have ∂s Vst [μ] ◦ st [μ] = ∂s ∂t st [μ] + ∇q ∂t st [μ]∂s Xst [μ], and by (3.18)

−1 ∂s X[μ] ◦ st [μ] = − ∇q st [μ] ∂s st [μ]. Thus, applying once more Lemma 3.13, we obtain the second inequality of the lemma. 2 t

Lemma 3.16. Let t0 ∈ [0, T ] and set σt0 = s0 [μ]# μ. We have: (i) tt0 [σt0 ] ◦ st0 [μ] = st [μ]. t

(ii) The maps tt0 [σt0 ] and t 0 [μ] are inverses of each other. (iii) Vst [μ] = Vtt0 [σt0 ]. (iv) ∂s st [μ] = −∇q st [μ]Vts [μ]. Proof. (i) Set S¯t = St ◦ St−1 , 0

where St = st [μ].

Obviously S¯t0 = Id,

(3.21)

 

−1 ∂t S¯0 = ∂t S0 ◦ St−1 = ∇ u ◦ S , S μ = ∇q u∗ S¯0 , S¯0 # σt0 . S q ∗ 0 0 # t 0 0

(3.22)

and

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We exploit the fact that St satisfies the second order differential equation in (3.14) to obtain

∂tt S¯t = −∇q F St ◦ St−1 , St # μ = −∇q F (S¯t , S¯t # σt0 ). 0

(3.23)

We combine (3.21), (3.22) and (3.23) and apply Remark 3.12 to conclude that S¯ is the unique fixed point of M[σt0 ]. In other words, S¯t = tt0 [σt0 ], which implies the desired conclusion. (ii) By the fact that [μ]ss = Id, (i) implies (ii). (iii) By (i) ∂t st [μ] = ∂t tt0 [σt0 ] ◦ st0 [μ]. But (i) allows us to compose the left hand-side of the identity with (st [μ])−1 and the right t hand-side with (s0 [μ])−1 ◦ (tt0 [σt0 ])−1 to obtain ∂t st [μ] ◦ (st [μ])−1 = ∂t tt0 [σt0 ] ◦ st0 [μ] ◦ (st0 [μ])−1 ◦ (tt0 [σt0 ])−1 = ∂t tt0 [σt0 ] ◦ (tt0 [σt0 ])−1 . This establishes (iii). (iv) By (i) and (ii) Id = st [μ] ◦ ts [σt ] and so, differentiating both sides of the identity with respect to s we obtain (iv). 2 Warning 3.17. It is worth pausing for the following remarks. (i) We would like to warn the reader that in Lemma 3.16(i) we are not making any claim about t the identity tt0 [ν] ◦ s0 [μ] = st [μ] for an arbitrary ν. Similarly, in Lemma 3.16(iii), no claim has been made about an identity as general as Vst [μ] = Vtt0 [ν] for an arbitrary ν. (ii) We have never attempted to write any identity linking elements of Os [μ] with those of Os¯ [μ] when s¯ = s. 4. Properties of  in the variables (t, s, q); continuity in μ Throughout this section we assume that T > 0, A > 0 satisfy (3.7). Definition 4.1. Let K := [0, T ] × [0, T ] × Td × P(Td ). We define the master map S : K → K by

S(t, s, q, μ) = t, s, st [μ](q), μ . Lemma 4.2. The following hold:

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S is continuous and S(·, ·, ·, μ) is 2A-Lipschitz. ∂t S, ∂tt S : K → Rd are continuous. The map S : K → K is a homeomorphism. V : K → Rd is continuous.

Proof. (i) Lemma 3.9 implies that S(·, ·, ·, μ) is 2A-Lipschitz. To complete the proof of (i) it suffices to show that if {μk }k ⊂ P(Td ) converges to μ, setting S k = (·, ·, ·, μk ), S = (·, ·, ·, μ) then {S k }k converges uniformly to S. By Lemma 3.9, S k is 2A-Lipschitz and so, {S k }k is equicontinuous. The Ascoli–Arzela lemma ensures the pre-compactness of the sequence in

C [0, T ] × [0, T ] × Td ; Td and so, the existence of a point of accumulation E. We invoke the continuity of M in all its variables as stated in Remark 3.4(ii) to conclude that E(·, s, ·) is a fixed point of Ms [μ] for every s. In other words, E(s, t, q) = st [μ](q). Thus there is a unique point of accumulation of {S k }k , and hence we conclude that the whole sequence {S k }k converges uniformly to S. (ii) By assumption ∇q F and ∇q u∗ are κ-Lipschitz. By (3.1), ∂t  is expressed in terms of . Similarly, by (3.2), ∂tt  is expressed in terms of . We use the continuity property of S to conclude that ∂t S and ∂tt S are continuous. (iii) By Lemma 3.16(ii), for any μ ∈ P(Td ), S(t, s, ·, μ) : Td → Td is bijective. It thus follows that S is a bijection which is continuous from the compact set K onto K. Hence, S is a homeomorphism. (iv) Recall that Vst [μ] = ∂t st [μ] ◦ Xst [μ] and so, by (ii) and (iii), V is continuous. 2 Lemma 4.3. The following functions are continuous and thus they are bounded: d×d . (i) ∇q  : K → R (ii) ∇q ∂t  : K → Rd×d . (iii) ∂s  : K → Rd .

Proof. (i) Let {sk }k , {tk }k ⊂ [0, T ], {qk }k ⊂ Td and {μk }k ⊂ P(Td ) be sequences converging respectively to s, t, q and μ. We are to show that {∇q stkk [μk ](qk )}k converges to ∇q [μ](t, s, q). By Theorem 3.8, [μk ](·, sk , ·) ∈ CA and so, for any q¯ ∈ Td   A|q¯ − q |2 k  tk  ¯ − stkk [μk ](qk ) − ∇q stkk [μk ](qk ) · (q¯ − qk ) ≤ sk [μk ](q) 2

(4.1)

and |∇q stkk [μk ](qk )| ≤ A. Hence {∇q stkk [μk ](qk )}k admits at least one point of accumulation, which we denote by P0 . Since, by Lemma 4.2(i),  is continuous, (4.1) implies   A|q¯ − q|2  t  ¯ − st [μ](q) − P0 · (q¯ − q) ≤ . s [μ](q) 2

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Thus, P0 = ∇q st [μ](q) is the unique point of accumulation. This proves (i). (ii) Since s [μ] is a fixed point of Ms [μ], (3.4) yields

∇tq st [μ] = ∇qq u∗ s0 [μ](q), s0 [μ]# μ ∇q s0 [μ](q) t −



∇qq F sτ [μ](q), sτ [μ]# μ ∇q sτ [μ](q)dτ.

(4.2)

0

Lemma 4.2(i) ensures the continuity of , while (i) of the current lemma ensures that ∇q  is continuous. Since, ∇qq u∗ is continuous, Remark 3.3 implies that

(t, s, q, μ) → ∇qq u∗ s0 [μ]q, s0 [μ]# μ is continuous. Similarly, we use the fact that ∇qq F is continuous to obtain that t (t, s, q, μ) →



∇qq F sτ [μ](q), sτ [μ]# μ ∇q sτ [μ](q)dτ

0

is continuous. Taking all these facts into consideration, representation formula (4.2) yields the continuity of ∇tq . (iii) Since, by Lemma 4.2(iv), V is continuous, and by (ii), ∇q  is continuous, the representation formula for ∂s st [μ] provided by Lemma 3.16(iv), ensures that ∂s  is continuous. 2 Lemma 4.4. The following maps are continuous and thus they are bounded: (i) X : K → Td . (ii) ∇q X : K → Rd×d . (iii) ∂t X : K → Rd . Proof. (i) Lemma 4.2(iii) gives that X is continuous on the compact set K. (ii) We use the representation formula (3.17), (i) and Lemma 4.3(i) to obtain (ii). (iii) By (3.18) ∂t Xst [μ] = −∇q Xst [μ]Vst [μ] and so, (ii) and Lemma 4.2(iv) yield (iii).

2

5. Minimality properties of  Throughout this section we assume that T > 0, A > 0 satisfy (3.7). The main result of this section is the following theorem.

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Theorem 5.1. Let s ∈ [0, T ], let μ ∈ P(Td ) and let σ ∈ AC2 (0, s; P(Td )) be a path of velocity v such that σs = μ. Then 1 − 3κCT A(s; σ, v) ≥ A(s; σ¯ , v¯ ) + 3T 2

s



W22 στ , σ¯ τ dτ,

(5.1)

0

where σ¯ t = st [μ]# μ,

v¯ t = Vst [μ].

As a consequence (σ¯ , v¯ ) is the unique minimizer of (6.2) which will be later considered in Section 6. Furthermore, for almost every t ∈ (0, s), Vst [μ] is the velocity of minimal norm for σ¯ and it belongs to Tσ¯ t P(Td ). Corollary 5.2. Let s ∈ [0, T ], let μ ∈ P(Td ) and as above set σ¯ t = st [μ]# μ,

v¯ t = Vst [μ].

Then:

(i) If r ∈ (0, T ] and σ ∈ AC2 0, r; P(Td ) has velocity v and σr = σ¯ r then A(r; σ, v) > A(r; σ¯ , v¯ ) unless σ = σ¯ . (ii) For every t ∈ [0, T ], Vst [μ] is the gradient of a function and so, it belongs to Tσ¯ t P(Td ), and ∇q Vst [μ] is a symmetric matrix. We postpone the proof of Theorem 5.1 and first derive Corollary 5.2 from Theorem 5.1. In Subsection 5.1 we will first show a discrete version of (5.1) and then use an approximation argument to prove Theorem 5.1 in its full generality in Subsection 5.2. Proof of Corollary 5.2. (i) Set σt∗ = rt [σr ]# σr ,

v∗t = Vrt [σr ].

By Theorem 5.1 A(r; σ, v) > A(r; σ ∗ , v∗ )

(5.2)

σt∗ = rt [σr ] ◦ sr [μ]# μ = st [μ]# μ = σ¯ t .

(5.3)

v∗t = v¯ t .

(5.4)

unless σ = σ ∗ . By Lemma 3.16(i)

By (iii) of the same lemma

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We have thus established that (σ¯ , v¯ ) = (σ ∗ , v∗ ). Using this in (5.2) we conclude the proof of (i). If we set r = T in the above argument, Theorem 5.1 also gives us that there is a set E ⊂ (0, T ) of full measure such that Vst [μ] belongs to Tσ¯ t P(Td ) for every t ∈ E . (ii) We divide the proof of (ii) into two steps. Step 1. Assume σs = s Ld and infTd s > 0. Since st [μ]# μ = σ¯ t , Lemma 3.13(i) implies that σ¯ t << Ld and so, there exists a nonnegative function ¯ t ≥ 0 such that σ¯ t = ¯ t Ld and

s (q) = ¯ t st [μ] det ∇q st [μ].

(5.5)

Since s [μ] ∈ CA , we use Remark 2.7 to obtain det ∇q s0 [μ] ≤ Ad /d d/2 ≤ Ad . Thus (5.5) implies 0<

1 inf s ≤ ¯ t . Ad Td

(5.6)

Therefore, if t ∈ E then there exists U¯ t ∈ W 1,2 (Td ) such that ∇ U¯ t = v¯ t ∈ W 2,∞ (Td )d (by Lemma 3.15) We thus have ∇vt = ∇ 2 U¯ t and so, ∇vt is symmetric. Any t ∈ [0, T ] can be written as the limit of a sequence {tn }n ⊂ E . Since {¯vtn }n converges uniformly to v¯ t and there exists U¯ tn ∈ W 1,2 (Td ) such that v¯ tn = ∇ U¯ tn we obtain a function U¯ t ∈ W 1,2 (Td ) such that v¯ t = ∇ U¯ t . Hence, vt belongs to Tσ¯ t P(Td ) and ∇vt is symmetric for all t ∈ [0, T ]. Step 2. Assume μ ∈ P(Td ) is arbitrary. Choose a sequence of positive probability densities n {T }n ⊂ C(Td ) such that infTd sn > 0 and lim W2 (μn , μ) = 0,

n→∞

where we have set μn = sn Ld . Set σtn = st [μn ]# μn ,

vnt = Vst [μn ].

Since, by Lemma 4.2, V is continuous, we conclude that {Vst [μn ]}n converges pointwise to Vst [μ] on Td . By Lemma 3.15 and the Sobolev Embedding Theorem, {Vs [μn ]}n is pre-compact in C 1 ([0, s] × Td )d and hence it converges to Vs [μ] in the C 1 -topology. Thus Vst [μ] is the gradient of a function U¯ t ∈ C 1 (Td ) and ∇q Vst [μ] is symmetric. 2 5.1. Optimality properties of discrete paths Let s ∈ (0, T ], let x¯1 , · · · , x¯n ∈ Td and define xi : [0, T ] → Td by xi (t) = st [μx¯ ](x¯i )

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Using (2.6) and (2.12), by the definition of  we have ⎧

x¨i = −∇μ F μx¯ x¯i (t) ⎨ (i) (ii) xi (s) = x¯i



⎩ (iii) x˙i (0) = ∇μ U∗ μx(0) xi (0) .

(5.7)

Let

y1 , · · · , yn ∈ W 1,2 0, T ; Td . Reordering and translating the y1 (s), · · · , yn (s) if necessary, we may assume that when t = s n n

1 1 W22 μx(s) , μy(s) = |xi (s) − yi (s)|2Td = |xi (s) − yi (s)|2 . n n i=1

i=1

Set 1 δ(xi (t),yi (t)) n n

γt =

i=1

so that

γs ∈ 0 μx(s) , μy(s) . We use the identity |y˙i |2 |x˙i |2 |y˙i − x˙i |2 1 = + + (y˙i − x˙i ) · x˙i 2n 2n 2n n and integrate by parts to obtain s

|y˙i |2 dt = 2n

0

s 

  x˙i s |x˙i |2 |y˙i − x˙i |2 1 . + − (yi − xi ) · x¨i dt + (yi − xi ) · 2n 2n n n 0

(5.8)

0

By (2.7) n n x 1 y

x

κ F μ ≤F μ + ∇μ F μ (xi ) · (yi − xi ) + |yi − xi |2 . n n i=1

i=1

Hence, using (5.7)(i), we conclude that s





s 

F μ dt ≤ 0

y

0

 n n 1 κ x¨i · (yi − xi ) + |yi − xi |2 dt. F μx − n n i=1

i=1

(5.9)

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Similarly, (2.13) and (5.7)(iii) imply n



1 U∗ μy(0) ≥ U∗ μx(0) + [x˙i (0) · (yi (0) − xi (0)) − κ|yi (0) − xi (0)|2 ]. n

(5.10)

i=1

Let v be a velocity for μx and let w be a velocity for μy . In fact wt is uniquely determined for almost all t ∈ (0, T ). We combine (5.8), (5.9) and (5.10) to conclude that 1 κ x˙i (s) · (yi (s) − xi (s)) − |yi (0) − xi (0)|2 n n n

A(s; μy , w) − A(s; μx , v) ≥

n

i=1

+

1 2n

i=1

n s 

 |y˙i − x˙i |2 − 2κ|yi − xi |2 dt.

(5.11)

i=1 0

Set t i (t) := yi (t) − xi (t) = yi (s) − xi (s) +

(y˙i (τ ) − x˙i (τ ))dτ. s

We have for 0 ≤ t ≤ s s |i (t)| ≤ |i (s)| +

˙ i (τ )|dτ |

0

and so, 3 |i (t)| ≤ 3|i (s)| + T 2 2

s

2

˙ i (τ )|2 dτ. |

0

This proves that 3 |i (0)| ≤ 3|i (s)| + T 2 2

s

2

˙ i (τ )|2 dτ |

(5.12)

0

and s

3 |i (t)| dt ≤ 3T |i (s)| + T 2 2 2

0

We combine (5.11), (5.12) and (5.13) to obtain

s

2

0

˙ i (τ )|2 dτ. |

(5.13)

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´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛ n n  1 1 − 3κCT ˙ i |2 dτ x˙i (s) · i (s) + | n 2n s

A(s; μy , w) − A(s; μx , v) ≥

i=1

−

i=1 0

κ (1 + 3T ) n

n

|i (s)|2 .

(5.14)

i=1

We use again (5.13) in (5.14) to obtain A(s; μy , w) − A(s; μx , v) n n  1 1 − 3κCT ≥ x˙i (s) · i (s) + |i |2 dτ n 3T 2 n s

i=1

−

i=1 0

n n

κ 1 − 3κCT |i (s)|2 − (1 + 3T ) |i (s)|2 . Tn n i=1

(5.15)

i=1

Hence,   n  1 y x A(s; μ , w) − A(s; μ , v) ≥ −  x˙i 2∞ n i=1

−

1 − 3κCT Tn

n

  n n s  1 − 3κCT 2  |i (s)| + |i |2 dτ 3T 2 n i=1

i=1 0

|i (s)|2 −

i=1

κ (1 + 3T ) n

n

|i (s)|2 .

(5.16)

i=1

We now use in (5.16) that s ∈ CA and

γt ∈ μx(t) , μy(t) ,



γs ∈ 0 μx(s) , μy(s) ,

|i (s)| = |i (s)|Td

to obtain



A(s; μy , w) − A(s; μx , v) ≥ −AW 2 μx(s) , μy(s) − BT W22 μx(s) , μy(s) 1 − 3κCT + 3T 2

s



W22 μx(τ ) , μy(τ ) dτ.

(5.17)

0

Here, BT is a constant depending only on T and κ. 5.2. Proof of Theorem 5.1 Proof of Theorem 5.1. For each integer m ≥ 1, let P m (Td ) denote the set of averages of m Dirac masses on Td . Let AC2 (0, s; P m (Td )) denote the set of pairs (σ, w) such that σ ∈ AC2 (0, s; P(Td )) and (σ, w) satisfy the following properties: There exist yi ∈ W 1,2 (0, s; Td ),

for i = 1, · · · , m

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such that 1 δyi (t) m m

σt =

and wt ◦ yi = y˙i ,

i = 1, · · · , m a.e.

i=1

Let s ∈ [0, T ], let μ ∈ P(Td ) and let σ ∈ AC2 (0, s; P(Td )) be a path of velocity w such that σs = μ. Proposition 5.1 of [21] provides us with a sequence of pairs (σ m , wm ) and a sequence of real numbers {r m }m ⊂ (0, 1) decreasing to 0 such that

sup t∈[0,s]

W2 (σtm , σt ) ≤ r m ,

1 2

s ||wm ||2σ m dt t

1 ≤ 2

0

s ||w||2σt dt + r m .

(5.18)

0

We combine (5.17) and (5.18) and use the fact that F and U∗ are κ-Lipschitz to obtain A(s; σ m , wm ) ≤ (1 + sκ)r m + κr m + A(s; σ, w).

(5.19)

d Let {xim (s)}m i=1 ⊂ T be such that

1 δxim (s) . m m

lim r¯ m = 0,

m→∞

where r¯ m = W2 (σ¯ sm , μ)

and σ¯ sm =

i=1

Set xim (t) = st [σ¯ sm ](xim (s)),

t m v¯ m ¯ s ], t = Vs [σ

σ¯ = st [μ]# μ,

v¯ t = Vst [μ].

Note that σ¯ tm = st [σ¯ sm ]# σ¯ sm

(5.20)

and thus t m ¯vm ¯ s ]σ¯ sm t σtm = ∂t s [σ

and

||¯vt ||σ¯ t = ||∂t st [μ]||μ .

(5.21)

Because {σ¯ sm }m converges to μ, and s [σ¯ sm ] ∈ CA we obtain that {s [σ¯ sm ]} is equicontinuous and so, {s [σ¯ sm ]} converges uniformly to s [μ] on [0, s] × Td . Hence, lim r m m→∞ 1

= 0,

where r1m = sup W22 (σ¯ tm , σ¯ t ).

(5.22)

t∈[0,s]

Similarly, {∂t s [σ¯ sm ]} converges uniformly to ∂t s [μ] on [0, s] × Td . Consequently, using the identities in (5.21), we have lim r2m = 0,

m→∞

    2 where r2m = sup ||¯vm vt ||2σ¯ t . t ||σtm − ||¯ t∈[0,s]

(5.23)

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Hence, since U∗ and F are Lipschitz, (5.22) and (5.23) imply lim A(s; σ¯ m , v¯ m ) = A(s; σ¯ , v¯ )

m→∞

(5.24)

We now apply inequality (5.17) to get



A(s; σ m , wm ) − A(s; σ¯ m , v¯ m ) ≥ −AW2 σsm , σ¯ sm − BT W22 (σsm , σ¯ sm 1 − 3κCT + 3T 2

s



W22 στm , σ¯ τm dτ.

(5.25)

0

Letting m tend to ∞ in (5.25), we use (5.18), (5.19), (5.22) and (5.24) to obtain A(s; σ, w) ≥ A(s; σ¯ , v¯ ) +

1 − 3κCT 3T 2

s



W22 στ , σ¯ τ dτ.

0

This concludes the proof of (5.1). A straightforward consequence of (5.1) is that (σ¯ , v¯ ) is the unique minimizer in (6.2). Denote by |σ¯  | the metric derivative of σ (see e.g. [4]). By Proposition 8.3.1 of [4] there exists a velocity v∗ for σ¯ such that for almost every t ∈ (0, s) ||v∗t ||σ¯ t ≤ |σ  |(t) ≤ ||¯vt ||σ¯ t .

(5.26)

This implies that all inequalities in (5.26) are equalities as otherwise we would get A(s; σ¯ , v∗ ) < A(s; σ¯ , v¯ ) which would contradict the minimality property of (σ¯ , v¯ ). By Proposition 8.4.5 of [4], since we have equalities in (5.26) for almost every t ∈ (0, s), we get v¯ t ∈ Tσ¯ t P(Td ) for almost every t ∈ (0, s). 2 Td ) 6. Hamilton–Jacobi equation on P(T Throughout this section we assume that T > 0, A > 0 satisfy (3.7). We assume (see Example 2.9) to be given U 0 ∈ C 3 (Td ), U 1 , φ ∈ C 3 (Td ) such that the latter two functions are even and the three functions satisfy ||φ||C 3 (Td ) ,

2||U 0 ||C 3 (Td ) ,

2||U 1 ||C 3 (Td ) ≤ κ.

We assume that for any q ∈ Td and any μ ∈ P(Td )  F (q, μ) = φ ∗ μ(q),

F(μ) = Td

1 φ ∗ μ(y)μ(dy), 2

(6.1)

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so that ∇q F (q, μ) = ∇φ ∗ μ(q). We set  u∗ (q, μ) = U 0 (q) + U 1 ∗ μ(q),

U∗ (μ) =



1 U 0 + U 1 ∗ μ (y)μ(dy). 2

Td

For s ∈ [0, T ], μ ∈ P(Td ) we define the value function  s  U(s, μ) = inf L(σ, v)dt + U∗ (σ0 ) | σs = μ ,

(6.2)

0

where the infimum is taken over the set of all pairs (σ, v) such that σ ∈ AC2 (0, s; P(Td )) and v is a velocity for σ . Recall that L is defined by (2.1). Using the terminology of [2] and [20], U is the unique metric viscosity solution to 

∂t U + H(μ, ∇μ U) = 0 in (0, T ) × P(Td ) U(0, ·) = U∗ on P(Td ).

(6.3)

Furthermore, U satisfies the semigroup property (the so-called dynamic programming principle): For any r ∈ [0, s] s  U(s, μ) = inf L(σ, v)ds + U (r, σr ) | σs = μ . r

Proposition 6.1. Fix s ∈ [0, T ], μ ∈ P(Td ) and set σ¯ t = st [μ]# μ,

v¯ t = Vst [μ],

∀ t ∈ [0, T ].

Then, for any r ∈ [0, T ], we have U(r, σ¯ r ) = A(r; σ¯ , v¯ ), in particular U (s, μ) = A(s; σ¯ , v¯ ). Proof. The result is a direct consequence of Corollary 5.2.

2

Remark 6.2. We recall that U is Lipschitz continuous on [0, T ] × P(Td ) (see [19,20]).

(6.4)

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6.1. Semiconvexity/semiconcavity properties of the value function Fix a positive integer n. For q = (q1 , · · · , qn ) ∈ (Td )n and p = (p1 , · · · , pn ) ∈ (Rd )n we define U0n (q) =

n n 1 0 1 1 U (qi ) + 2 U (qi − qj ) n 2n i,j =1

i=1

and n |p|2 1 φ(qi − qj ). − 2 2n 2n

Ln (q, p) =

i,j =1

We notice that for any q ∈ (Td )n , p ∈ (Rd )n −∇q F (qi , μq ) = n∇qi Ln (q, p), ∇q u∗ (qi , μ

q

) = n∇qi U0n (q).

(6.5) (6.6)

For x = (x1 , · · · , xn ) ∈ (Td )n we define U (s, x) = inf n

s

y

 Ln (y, y˙ )dt + U0n (y(0)) | y(0) = x .

0

Proposition 6.1 implies that U n (s, x) = U(s, μx ). Lemma 6.3. Let s ∈ (0, T ), let x = (x1 , · · · , xn ) ∈ (Td )n , x∗ = (x1∗ , · · · , xn∗ ) ∈ (Td )n . Then there ∗

exists γ ∈ 0 μx , μx such that (i)

∗

U s, μx ≤ U s, μx +

 Vss [μx ](q) · (b − q)γ (dq, db)

Td ×Td

∗

+ κ(1 + s)W22 μx , μx .

(6.7)

(ii)

∗

U s, μx ≥ U s, μx +

 Vss [μx ](q) · (b − q)γ (dq, db)

Td ×Td

 1  1 ∗

+ κ(T − s) . − W22 μx , μx 2 (T − s)

(6.8)

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Proof. (i) For any t ∈ [0, T ] we set zi (t) = st [μx ](xi ),

i = 1, · · · , n.

Then Proposition 6.1 yields s Ln (z, z˙ )dt + U0n (z(0)).

U (s, x) = U (s, μ ) = x

n

(6.9)

0

Since

z˙ i (0) = ∂t s0 [μx ](xi ) = ∇q u∗ zi (0), μz(0) ,

z¨i (t) = −∇q F (zi (t), μz(t) ),

(6.5) and (6.6) imply ∇U0n (z(0)) =

z˙ (0) , n

∇qi Ln (z(t), z˙ (t)) =

z¨ (t) . n

(6.10)

It follows from (2.7) and (2.13) that the functions U0n and Ln (·, p) for every p ∈ (Rd )n are 2κ/n-concave. We now set yi (t) = zi (t) + xi∗ − xi . We have ∗

s Ln (y, y˙ )dt + U0n (y(0)).

U (s, x ) ≤ n

(6.11)

0

Hence, using (6.9), (6.10), (6.11) and the semiconcavity of U0n , Ln (·, z˙ ), we obtain ∗

s [Ln (y, y˙ ) − Ln (z, z˙ )]dt + U0n (y(0)) − U0n (z(0))

U (s, x ) ≤ U (s, x) + n

n

0

s ≤ U n (s, x) +

z¨ κs z˙ (0) κ · (x∗ − x)dt + |x∗ − x|2 + · (x∗ − x) + |x∗ − x|2 n n n n

0

= U n (s, x) +

z˙ (s) κ(1 + s) ∗ · (x∗ − x) + |x − x|2 . n n

(6.12)

Reordering the points x1∗ , · · · , xn∗ and translating them if necessary, we may assume without loss of generality that 1 ∗

|xi − xi∗ |2 = W22 μx , μx n n

i=1

(6.13)

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and thus 1 ∗

δ(xi ,xi∗ ) ∈ 0 μx , μx . n n

i=1

Therefore (6.12) and (6.13) imply (6.7). (ii) By Proposition 6.1 T

T Ln z, z˙ )dt

U (T , z(T )) = n

Ln (z, z˙ )dt + U n (s, x).

+ U0n (z(0)) =

(6.14)

s

0

We set yi (t) = zi (t) +

T −t ∗ (x − xi ), T −s i

t ∈ [s, T ].

Since y(s) = x∗ and y(T ) = z(T ) we use the fact that U satisfies the semigroup property (6.4) to obtain T

∗

Ln (y, y˙ )dt.

U (s, x ) ≥ U (T , z(T )) − n

n

s

This, together with (6.14), implies T

∗

U (s, x ) ≥ U (s, x) + n

n



Ln (z, z˙ ) − Ln (y, y˙ ) dt.

s

Therefore, by the semiconcavity of Ln (·, z˙ ) and (6.10), we obtain 1 |x∗ − x|2 1 U (s, x ) ≥ U (s, x) − + 2n T − s n n

∗

T

n

−

1 n

T

z¨ · (x∗ − x)

s

≥ U n (s, x) +

s

z˙ · (x∗ − x) dt T −s

T −s T −t κ dt − |x∗ − x|2 T −s n 3

 z˙ (s) · (x ∗ − x) |x ∗ − x|2  1 − + κ(T − s) . n 2n T −s

We conclude the proof arguing as in part (i). 2

Theorem 6.4. Let μ, μ∗ ∈ Td and let γ ∈ 0 μ, μ∗ .

(6.15)

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(i) If s ∈ [0, T ] then

∗

U s, μ





≤ U s, μ +

 Vss [μ](q) · (b − q)γ (dq, db)

Td ×Td



+ κ(1 + s)W22 μ, μ∗ .

(6.16)

(ii) If s ∈ [0, T )

∗

U s, μ





≥ U s, μ +

 Vss [μ](q) · (b − q)γ (dq, db)

Td ×Td



 1 1 + κ(T − s) . − W22 μ, μ∗ 2 (T − s)

(6.17)

(iii) For any s ∈ (0, T ) and t ∈ [0, T ) we have ∇μ U(t, σ¯ t ) = Vst [μ], where σ¯ t = st [μ]# μ. Proof. (i) The function U is continuous by Remark 6.2. Since, by Lemma 4.2, V is continuous, it suffices to prove (6.16) for s ∈ (0, T ). Using again the fact that U and V are continuous, since every μ and μ∗ can be approximated by averages of Dirac masses, it follows from Lemma 6.3

that there exists γ ∈ 0 μ, μ∗ such that (i) holds. It remains to show that (i) holds for all γ ∈

0 μ, μ∗ . We fix such γ , and let μ∗λ be the geodesic defined by 

ϕ(q)μ∗λ (dq) =

Td





ϕ (1 − λ)q + λb γ (dq, db).

Td ×Td



By Lemma 7.2.1 of [4], for λ ∈ (0, 1), 0 μ, μ∗λ contains a unique element γλ . Thus, U (s, μ∗λ ) ≤ U (s, μ) +







Vss [μ](q) · (b − q)γλ (dq, db) + κW22 μ, μ∗λ 1 + s .

Td ×Td

Letting λ tend to 1 we obtain (6.16). Similar arguments yield (6.17). (iii) By (i), Vss [μ] belongs to ∂ · U(s, μ) ∩ ∂· U (s, μ), whereas Corollary 5.2 ensures that s Vs [μ] ∈ Tμ P(Td ). Therefore, by the remark in Definition 2.5(iii), ∇μ U (s, μ) = Vss [μ]. Using Lemma 3.16(iii) we now have ∇μ U(t, σ¯ t ) = Vtt [σ¯ t ] = Vst [μ].

2

Remark 6.5. Lemma 6.3 and Theorem 6.4 correct and sharpen the statements of Theorem 5.1 (iii) and Theorem 5.2 (iv) of [19].

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Lemma 6.6. Let s ∈ (0, T ], let μ ∈ P(Td ) and set vt = Vst [μ],

σt = st [μ]# μ.

Then: (i) (σ, v) satisfies the following system of equations, where the first identity in (6.18) holds pointwise, 

∂t v + ∇vv = −∇q F (·, σt ) v0 = ∇μ U∗ [σ0 ] = ∇U 0 + ∇U 1 ∗ σ0 .

(6.18)

(ii) ∇vt is the gradient of a function and thus it is a symmetric matrix for any t ∈ [0, T ]. Proof. Observe first that by Lemma 3.15 and the Sobolev Embedding Theorem, v is continuously differentiable on (0, T ) × Td . (i) By the definition of v we have ∂t st [μ] = vt (st [μ]) and so, differentiating with respect to t and using the first equation in (3.14) we obtain   −∇q F st [μ](q), st [μ]# μ = ∂tt st [μ](q)



= ∂t vt st [μ]q + ∇vt st [μ]q ∂t st [μ](q). Thus   −∇q F q, st [μ]# μ = ∂t vt (q) + ∇vt (q)vt (q), which gives the first identity in (6.18). The second identity in (6.18) follows from Theorem 6.4(iii). Part (ii) is already stated in Corollary 5.2. 2 7. Weak solution to the first order Mean Field equations Throughout this section we assume that T > 0, A > 0 satisfy (3.7). We also assume that F , u∗ and U∗ are given through functions φ, U 0 , and U 1 satisfying the assumptions imposed in Section 6. Given s ∈ [0, T ], q ∈ Td , μ ∈ P(Td ), we define

u(s, q, μ) = u∗ q, s0 [μ]# μ



s  −



|Vsτ [μ](q)|2 + F q, sτ [μ]# μ dτ, 2

(7.1)

0

and set σ¯ t = st [μ]# μ and

v¯ t = Vst [μ]

∀ t ∈ [0, T ].

(7.2)

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Since, by Lemma 3.16, Vtτ [σ¯ t ] = Vsτ [μ] and

tτ [σ¯ t ]# σ¯ t = tτ [σ¯ ] ◦ st [μ]# μ = sτ [μ]# μ = σ¯ τ ,

we conclude that t  u(t, q, σ¯ t ) = u∗ (q, σ¯ 0 ) −



|Vsτ [μ](q)|2 + F q, sτ [μ]# μ dτ, 2

∀t ∈ [0, T ].

(7.3)

0

Lemma 7.1. We have, for every s ∈ [0, T ], μ ∈ P(Td ): (i) For any (t, q) ∈ (0, T ) u(0, ·, μ) = u∗ (·, μ)

and ∇q u(t, q, σ¯ t ) = v¯ t (q) = ∇μ Ut (t, σ¯ t )(q).

(ii) t → u t, q, σ¯ t is continuously differentiable and

|∇q u(t, q, σ¯ t )|2 ∂t u(t, q, σ¯ t ) + + F (q, σ¯ t ) = 0, 2

∀ (t, q) ∈ (0, T ) × Td .

Proof. (i) The identity u(0, ·, μ) = u∗ (·, μ) is straightforward to check. We substitute Vsτ [μ] by v¯ τ in (7.3) and differentiate the subsequent identity with respect to q to obtain

∇q u(t, q, σ¯ t ) = ∇q u∗ (q, σ¯ 0 ) −

t 

 ∇ T v¯ τ (q)¯vτ (q) + ∇q F (q, σ¯ τ ) dτ.

0

We use that, by Lemma 6.6(ii), ∇ v¯ τ is symmetric and then use Lemma 6.6(i) to conclude that

∇q u(t, q, σ¯ t ) = ∇q u∗ (q, σ¯ 0 ) −

t   ∇ v¯ τ (q)¯vs (q) + ∇q F (·, σ¯ τ ) dτ 0

t ∂t v¯ τ (q)dτ.

= ∇q u∗ (q, σ¯ 0 ) + 0

We combine this with the fact that, by Lemma 6.6(i), ∇q u∗ (q, σ¯ 0 ) = v¯ 0 , and use Theorem 6.4(iii) to obtain ∇q u(t, q, σ¯ t ) = ∇q u∗ (q, σ¯ 0 ) + v¯ t (q) − v¯ 0 (q) = v¯ t (q) = ∇μ U (t, σ¯ t )(q).

(7.4)

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(ii) By Lemma 4.2(iv), V is continuous in all its variables. Since φ and  are continuous we conclude that for every y, 



φ q − sτ [μ](y) μ(dy)

τ → F (q, σ¯ τ ) = Td

is continuous. Using the representation formula provided by (7.3) we thus conclude that the

function t → u t, q, σ¯ t is continuously differentiable and

|V t [μ](q)|2 ∂t u(t, q, σ¯ t ) + s + F (q, σ¯ t ) = 0. 2 We now substitute Vst [μ] by ∇q u(t, q, σ¯ t ) to conclude the proof of (ii).

2

We now fix s and μ and define the function U (which depends on s and μ) by

U (t, q) ≡ Us,μ (t, q) = u t, q, st [μ]# μ so that, by (7.3),

U (t, q) = u∗ (q, σ¯ 0 ) −

t 

 |¯vτ (q)|2 + F (q, σ¯ τ ) dτ. 2

0

Corollary 7.2. The following hold: (i)

U ∈ W 2,∞ (0, T ) × Td . (ii) U is a classical solution (hence also a viscosity solution) to (7.5)(a), where ⎧ ⎪ ⎨ (a) (b) ⎪ ⎩ (c)

+ F (q, σ¯ t ) = 0 ∂t U (t, q) + |∇U (t,q)| 2

∂ σ¯ t + ∇ · (σ¯ t ∇U ) = 0 in D (0, T )) × Td U (0, q) = u∗ (q, σ¯ 0 ), σ¯ s = μ. 2

(7.5)

Moreover, if μ has a density with respect to the Lebesgue measure, then so does σ¯ t for every t ∈ [0, T ]. Proof. By Lemma 7.1, (7.5)(a) holds in the classical sense and ∂t U (t, q) = −

|Vst [μ](q)|2 − 2

 Td



φ q − sτ [μ](y) μ(dy),

∇q U (t, q) = Vst [μ](q).

(7.6)

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d Lemma 3.15 guarantees that Vs [μ] belongs to W 2,∞ (0, T ) × Td , while Theorem 3.8(ii) en

sures that s [μ] belongs to W 2,∞ (0, T ) × Td ; Td . Hence, by (7.6),

U ∈ W 2,∞ (0, T )) × Td and, using the fact that v is a velocity for σ and that ∇q Ut = v¯ t , we obtain (7.5)(b). The two identities in (7.5)(c) follow from Lemma 7.1. It is clear from the definition of σ¯ t and the regularity of Xst [μ] (the inverse of st [μ]) given by Lemma 3.13(iii), that if μ has a density with respect to the Lebesgue measure, then so does σ¯ t for every t ∈ [0, T ]. 2 We notice that if u(·, q, ·) is regular enough then, by Lemma 9.8,

∂t u(t, q, σ¯ t ) = ∂t u(t, q, σ¯ t ) +

 ∇μ u(t, q, σ¯ t )(z) · v¯ t σ¯ t (dz).

(7.7)

Td

When t = s then σ¯ t = μ and so, we may then use Lemma 7.1(i) to substitute ∇q u(s, z, μ) for v¯ s (z) in (7.7) and then use Lemma 7.1(ii) to obtain  ∂s u(s, q, μ) +

∇μ u(s, q, μ)(z) · ∇q u(s, z, μ) μ(dz) +

|∇q u(s, q, μ)|2 + F (q, μ) = 0. 2

Td

Thus to prove that u is a pointwise (strong) solution to the master equation (1.1), it suffices to show that u is regular enough in all its variables. The above comments suggest the following definition of a weak solutions to (1.1). Let u : [0, T ] × Td × P(Td ) → R be a continuous function such that u(0, ·, ·) = u∗ . Definition 7.3. We say that u is a weak solution

to (1.1) if for every s ∈ (0, T ) and every μ ∈ P(Td ) there exists a path σ ∈ AC2 0, T ; P(Td ) with a velocity v such that the following hold: (i) For almost every t ∈ (0, T ), ∇q u(t, ·, σt ) exists σt -almost everywhere. (ii) σs = μ and for almost every t ∈ (0, T ) vt = ∇q u(t, ·, σt ),

σt –almost everywhere.

(iii) Uμ (t, q) := u(t, q, σt ) is a viscosity solution to (7.5)(a). 8. Regularity properties of (t, s, q, ·); a discretization approach Throughout this section we assume that T > 0, A > 0 satisfy (3.7) and that F , u∗ and U∗ are given through functions φ, U 0 , and U 1 satisfying the assumptions imposed in Section 6. We recall that  and S are given by Definitions 3.10 and 4.1.

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Remark 8.1. Let {a k }k ⊂ [0, ∞) be a sequence and let α and β < 1 be two nonnegative numbers such that a k ≤ α + βa k−1 for all natural numbers k. Then ak ≤ α

k−1

β i + β k a0 ≤

i=1

α + a0. 1−β

8.1. Spatial derivatives of the discrete master map Throughout this subsection n is a fixed natural number. To x = (x1 , · · · , xn ) ∈ (Td )n we associate the measure 1 δxi . n n

μx =

i=1

For s ∈ [0, T ] recall that Ms [μ] is the map defined in (2.17). For x ∈ (Td )n and any continuous map S : [0, T ] × [0, T ] × Td × (Td )n → Td , we define Ms [x](S) := Ms [μx ](S(·, s, ·, x)), i.e.

Ms [x](S)(t, q) = q + (t − s)∇U 0 S(0, s, q, x)

1 ∇U 1 S(0, s, q, x) − S(0, s, xj , x) n n

+ (t − s)

j =1

1 n n

+

s

s ds

j =1 t



∇φ S(τ, s, q, x) − S(τ, s, xj , x) dτ.

(8.1)

0

Corollary 8.2. Let S 0 (t, s, q, x) ≡ q. Defining inductively

S k (t, s, q, x) = Ms [x] S k−1 (t, q), the following hold: (i) S k (·, ·, ·, x) ∈ CA∗ for every x ∈ (Td )n . (ii) There is a constant CA independent of k such that for any i ∈ {1, · · · , n} ∇xi S k ∞ ,

∇xi ∂t S k ∞ ≤

CA . n

(8.2)

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Proof. Since S 0 (·, ·, ·, x) ∈ CA∗ , Lemma 3.9 implies (i). (ii) We have

∇xi S k (t, s, q, x)



= (t − s)∇ 2 U 0 S k−1 (0, s, q, x) ∇xi S k−1 (0, s, q, x)

1 − (t − s)∇ 2 U 1 S k−1 (0, s, q, x) − S k−1 (0, s, xi , x) ∇q S k−1 (0, s, xi , x) n n



1 + (t − s) ∇ 2 U 1 S k−1 (0, s, q, x) − S k−1 (0, s, xi , x) i (0, q, xj , x) n j =1

1 − n

s

l dl

t

  ∇ 2 φ S k−1 (τ, s, q, x) − S k−1 (τ, s, xi , x) ∇q S k−1 (τ, s, xi , x)dτ

0

1 + n n

s

l dl

j =1 t

  ∇ 2 φ S k−1 (τ, s, q, x) − S k−1 (τ, s, xj , x) i (τ, q, xj , x)dτ,

(8.3)

0

where we do not display the s dependence in i (τ, q, xj , x) := ∇xi S k−1 (τ, s, q, x) − ∇xi S k−1 (τ, s, xj , x). We exploit (8.3) to infer ∇xi S k ∞ ≤

κCT ∇q S k−1 ∞ + κT (2 + T )∇xi S k−1 ∞ . 2n

Since S k−1 (·, ·, ·, x) ∈ CA∗ , we conclude that ∇xi S k ∞ ≤

κACT + 2κCT ∇xi S k−1 ∞ . 2n

(8.4)

We apply Remark 8.1 to (8.4) and use the fact that ∇xi S 0 ≡ 0 to obtain ∇xi S k ∞ ≤

κACT . 2n(1 − 2κCT )

Direct differentiation yields

∇xi ∂t S k (t, s, q, x)

= ∇ 2 U 0 S k−1 (0, s, q, x) ∇xi S k−1 (0, s, q, x)

1 − ∇ 2 U 1 S k−1 (0, s, q, x) − S k−1 (0, s, xi , x) ∇q S k−1 (0, s, xi , x) n

(8.5)

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1 2 1 k−1 ∇ U S (0, s, q, x) − S k−1 (0, s, xi , x) i (0, q, xj , x) n n

+

j =1

1 + n

t



∇ 2 φ S k−1 (τ, s, q, x) − S k−1 (τ, s, xi , x) ∇q S k−1 (τ, s, xi , x)dτ

0

1 − n n

t



∇ 2 φ S k−1 (τ, s, q, x) − S k−1 (τ, s, xj , x) i (τ, q, xj , x)dτ.

j =1 0

Estimating we thus obtain ∇xi ∂t S k ∞ ≤ 2κ(1 + T )∇xi S k−1 ∞ +

κA(1 + T ) . n

This, together with (8.5), yields ∇xi ∂t S k ∞ ≤ 2κ(1 + T )

κACT κA(1 + T ) + . 2n(1 − 2κCT ) n

We can choose CA in terms of T , κ and A to conclude the proof of the lemma. 2 Corollary 8.3. The sequence {S k }k defined in Corollary 8.2 converges uniformly on [0, T ] × [0, T ] × Td × (Td )n to the function S defined by S(t, s, q, x) = st [μx ](q). Furthermore, the following hold: (i) S satisfies (8.2), S(·, ·, ·, x) ∈ CA∗ for every x ∈ (Td )n , and S(t, s, q, x) = S(t, s, q, x¯ ) if x¯ is a permutation of x. (ii) Increasing the value of CA if necessary we have: ∇qxi S∞ ≤ ∇xj xi S∞ ≤

CA , n

CA , n2

∇xi xi S∞ ≤

i = 1, . . . , n.

(8.6)

i, j = 1, . . . , n, i = j.

(8.7)

CA , n

i = 1, . . . , n.

(8.8)

Proof. By Lemma 3.9, for each x fixed, {S k (·, ·, ·, x)}k converges uniformly to S(·, ·, ·) defined by S(t, s, q, x) = st [μx ](q). Thanks to the Arzela–Ascoli Lemma, the bounds on {S k }k and its derivatives provided by Corollary 8.2 imply that {S k }k converges uniformly to S on [0, T ] × [0, T ] × Td × (Td )n . (i) The fact that S satisfies (8.2) and S(·, ·, ·, x) ∈ CA∗ follows from Corollary 8.2. Since for t, s and q fixed, S(t, s, q, ·) depends only on μx then S(t, s, q, x) = S(t, s, q, x¯ ) if x¯ is a permutation of x. (ii) We differentiate both sides of (8.3) with respect to q to obtain an identity from which we derive the upper bound

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  κC T ∇qxi S k ∞ ≤ κCT 2∇q S k−1 ∞ ∇xi S k−1 ∞ + ∇q,xi S k−1 ∞ + ∇q S k−1 2∞ . 2n This, together with Corollary 8.2 implies ∇qxi S k ∞ ≤ κCT ∇q,xi S k−1 ∞ +

2κACA CT κA2 CT + . n n

We apply Remark 8.1 and use the fact that ∇qxi S 0 ∞ = 0 and then replace CA by an appropriate larger constant, still denoted by CA , such that ∇qxi S k ∞ ≤

CA . n

Letting k tend to ∞ we obtain (8.6). For (8.7) we differentiate both sides of (8.3) with respect to xj , j = i, and estimate the subsequent expression to obtain ∇xj xi S k ∞ ≤ 2κCT ∇xj xi S k−1 ∞ + A1 ∇xi S k−1 ∞ · ∇xj S k−1 ∞  A2  + ∇xj S k−1 ∞ · ∇q S k−1 ∞ + ∇xj q S k−1 ∞ , n for some A1 , A2 depending only on κ, T . We then use (i), (8.2) and (8.6) to get ∇xj xi S k ∞ ≤ 2κCT ∇xj xi S k−1 ∞ +

A3 n2

for some A3 depending only on κ, T , A, CA . We now apply Remark 8.1 and use the fact that ∇xl xi S 0 ∞ = 0 and to obtain a constant that we still denote by CA such that ∇xj xi S k ∞ ≤

CA . n2

Letting k tend to ∞ yields (8.7). To obtain (8.8) we differentiate both sides of (8.3) with respect to xi and repeat similar arguments. However now the differentiation of the second and fourth lines in (8.3) will produce terms that can only be bounded by  C ∇q S k−1 2∞ + ∇qq S k−1 ∞ n for some constant C. This is the reason why we obtain a weaker estimate than (8.7). 2 Corollary 8.4. Let s ∈ (0, T ), t¯, tˆ ∈ [0, T ], q, ¯ qˆ ∈ Td , x¯ , xˆ ∈ (Td )n

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be such that n

1 W22 μx¯ , μxˆ = |xˆi − x¯i |2Td . n i=1

Set S¯ = S(t¯, s, q, ¯ x¯ ), Sˆ = S(tˆ, s, q, ˆ xˆ ),

(8.9)

∂t S¯ = ∂t S(t¯, s, q, ¯ x¯ ), ∇q S¯ = ∇q S(t¯, s, q, ¯ x¯ ), ∇xi S¯ = ∇xi S((t¯, s, q, ¯ x¯ ).

(8.10)

and

Then there is a constant, still denoted by CA , such that ¯ qˆ − q) |Sˆ − S¯ − (tˆ − t¯)∂t S¯ − ∇q S( ¯ −

n

¯ xˆi − x¯i )| ∇xi S(

i=1



 ¯ 2Td + W22 μx¯ , μxˆ . ≤ CA (tˆ − t¯)2 + |qˆ − q|

(8.11)

Proof. We write the Taylor expansion of S around (t¯, q, ¯ x¯ ) to obtain that the expression in the left hand side of (8.11) is bounded by L + K = L + K1 + K2 , where L=

  n

1 |xˆi − x¯i |Td · |xˆj − x¯j |Td ∇xi xj S∞ + |qˆ − q| ¯ 2Td ∇qq S∞ , (tˆ − t¯)2 ||∂tt S||∞ + 2 i,j =1

¯ + K1 = ∇q ∂t S∞ |tˆ − t¯| · |qˆ − q|

n

∇xi ∂t S∞ |tˆ − t¯| · |xˆi − x¯i |Td

i=1

and K2 =

n

∇qxi S∞ |qˆ − q| ¯ Td · |xˆi − x¯i |Td .

i=1

Corollaries 8.2 and 8.3 provide us with upper bounds on the partial derivatives of S up to the second order. These bounds yield

i=j

|xˆi − x¯i |Td · |xˆj − x¯j |Td ∇xi xj S∞ ≤ ≤

 CA  |xˆi − x¯i |2Td + |xˆj − x¯j |2Td 2 2n i=j

n CA |xˆi − x¯i |2Td n i=1

(8.12)

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6617

and n

|xˆi − x¯i |2Td ∇xi xi S∞ ≤

i=1

n CA |xˆi − x¯i |2Td . n

(8.13)

i=1

We also have (tˆ − t¯)2 ∂tt S∞ + |qˆ − q| ¯ 2Td ∇qq S∞ ≤ κ(t˜ − t¯)2 + A|q˜ − q| ¯ 2Td .

(8.14)

We combine (8.12), (8.13) and (8.14) to conclude that 

 L ≤ C˜ A (tˆ − t¯)2 + |qˆ − q| ¯ 2Td + W22 μx¯ , μxˆ

(8.15)

for some constant C˜ A . Since ∇xi ∂t S∞ |tˆ − t¯| · |xˆi − x¯i |Td ≤

2 2 CA |tˆ − t¯| + |xˆi − x¯i |Td , n 2

summing up we have n

∇xi ∂t S∞ |tˆ − t¯| · |xˆi − x¯i |Td ≤

i=1

n CA CA |xˆi − x¯i |2Td . |tˆ − t¯|2 + 2 2n

(8.16)

i=1

Similarly, n

∇qxi S∞ |qˆ − q| ¯ Td · |xˆi − x¯i |Td ≤

i=1

n CA CA |xˆi − x¯i |2Td . |qˆ − q| ¯ 2Td + 2 2n

(8.17)

i=1

Notice also that ∇q ∂t S∞ |tˆ − t¯| · |qˆ − q| ¯ ≤

 3κA  ¯ 2Td . |tˆ − t¯|2 + |qˆ − q| 4

We combine (8.16), (8.17) and (8.18) to conclude that 

 K ≤ D˜ A (tˆ − t¯)2 + |qˆ − q| ¯ 2Td + W22 μx¯ , μxˆ for some constant D˜ A . This, together with (8.15), completes the proof of the lemma. 2 Corollary 8.5. Increasing the value of CA if necessary we have: (i) ∂ttt S∞ , ∇q ∂tt S∞ , ∇qq ∂t S∞ ≤ CA .

(8.18)

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(ii) For i = 1, . . . , n ∇xi ∂tt S∞ , ∇q,xi ∂t S∞ ≤

CA . n

(iii) For i, j = 1, . . . , n, i = j 1 ∇xi xi ∂t S∞ , n

∇xi xj ∂t S∞ ≤

CA . n2

Proof. (i) Since ∂tt S(t, s, q, x) = −∇q F (S(t, s, q, x), S(t, s, ·, x)# μx )

1 ∇φ S(t, s, q, x) − S(t, s, xj , x) , n n

=−

(8.19)

j =1

we have



1 2 ∇ φ S(t, s, q, x) − S(t, s, xj , x) ∂t S(t, s, q, x) − ∂t S(t, s, xj , x) . n n

∂ttt S(t, s, q, x) = −

j =1

We use the fact that S(·, ·, ·, x) ∈ CA∗ (see Corollary 8.3(i)) to conclude that ∂ttt S∞ ≤ 2κA. Similarly, we obtain the remaining inequalities in (i). (ii) Differentiating both sides of (8.19) with respect to xi we obtain ∇xi ∂tt S(t, s, q, x)

1 = ∇ 2 φ S(t, s, q, x) − S(t, s, xi , x) ∇q S(t, s, xi , x) n n



1 2 ∇ φ S(t, s, q, x) − S(t, s, xj , x) ∇xi S(t, s, q, x) − ∇xi S(t, s, xj , x) . − n j =1

This, together with Corollary 8.3(i), gives us ∇xi ∂tt S∞ ≤

κA 2κCA + . n n

The other inequality in (ii) is proved similarly. (iii) We differentiate ∇xj ∂t S with respect to xi and use Corollary 8.3.

2

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Corollary 8.6. Let t¯, tˆ ∈ [0, T ], q, ¯ qˆ ∈ Td , x¯ , xˆ ∈ (Td )n be such that n

1 W22 μx¯ , μxˆ = |xˆi − x¯i |2Td . n i=1

Using the notation (8.9) and (8.10), increasing the value of CA if necessary, we have ¯ qˆ − q) |∂t Sˆ − ∂t S¯ − (tˆ − t¯)∂tt S¯ − ∇q ∂t S( ¯ −

n

¯ xˆi − x¯i )| ∇xi ∂t S(

i=1



 ¯ 2Td + W22 μx¯ , μxˆ . ≤ CA (tˆ − t¯)2 + |qˆ − q|

(8.20)

Proof. We exploit Corollary 8.5 to complete the proof in exactly the same way Corollaries 8.2 and 8.3 are used to prove Corollary 8.4. 2 8.2. Spatial derivatives of the inverse of the master map Throughout this section n is a fixed natural number. We define S(t, s, q, x) := (t, s, st [μx ](q), x) = (t, s, S(t, s, q, x), x), R(t, s, b, x) := Xst [μx ](b). By Corollary 8.3, for every s ∈ [0, T ],   S(·, s, ·,·) ∈ W 2,∞ (0, T ) × Td × (Td )n ; (0, T ) × Td × (Td )n . Denote by I ∈ Rd×d the identity matrix. We exploit Lemma 4.2(iii) to infer that S is a homeomorphism and we denote its inverse by X . Observe that

X (t, s, b, x) = t, s, R(t, s, b, x), x . Denoting the null d × d matrix by 0 and the d × 1 null matrix by 0¯ we have ⎡

1 0···0 ⎢ 0¯ ∇q S ⎢ ⎢ 0¯ 0 ⎢ ∇(t,q,x) S = ⎢ ⎢ 0¯ 0 ⎢ ⎢ .. .. ⎣. . ¯0 0

0···0 0···0 ∇x1 S ∇x2 S I 0 0 0

I .. . 0

··· ··· ··· ··· I ···

⎤ 0···0 ∇xn S ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎦ I

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6620

and so, exploiting Lemma 3.13(i), we obtain 1 det ∇(t,q,x) S = det ∇q S ≥ . 4

(8.21)

We use the Inverse Function Theorem to conclude that for every s ∈ [0, T ]   X (·, s, ·,·) ∈ W 2,∞ (0, T ) × Td × (Td )n ; (0, T ) × Td × (Td )n .

(8.22)

Denote by adj(A) the adjugate of a d × d matrix A so that adj(A)A = (det A)I. We have adj(∇q S) ◦ R, det ∇q S  adj ∇ S  q ∂t S ◦ R, ∂t R = − det ∇q S ∇q R =

(8.23) (8.24)

and ∇xi R = −

adj(∇q S)∇xi S ◦ R. det ∇q S

(8.25)

Theorem 8.7. There exists a constant D¯ A independent of n such that the following hold: (i) ||∂s R||∞ ,

||∂t R||∞ ,

||∇q R||∞ ,

||∂tt R||∞ ,

||∇qq R||∞ ,

||∂t ∇q R||∞ ≤ D¯ A .

(ii) For i = 1, . . . , n ||∇xi R||∞ ,

||∇xi q R||∞ ,

||∇xi xi R||∞ ,

||∂t ∇xi R||∞ ≤

D¯ A . n

(iii) For i, j = 1, . . . , n, i = j ||∇xi xj R||∞ ≤

D¯ A . n2

Proof. We obtain (i) as a consequence of Lemmas 3.13 and 4.4. We differentiate the expression in (8.25) successively with respect to q, xj , t, use the fact that R(t, s, ·, x) and S(t, s, ·, x) are inverses of each other, and then we use the bounds obtained in Corollary 8.3 to conclude that (ii) and (iii) hold. 2

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Remark 8.8. Denoting V (t, s, b, x) := Vst [μx ](b), we have ∇xi V (t, s, b, x) = ∇q ∂t S(t, s, R(t, s, b, x), x)∇xi R(t, s, b, x) + ∇xi ∂t S(t, s, R(t, s, b, x), x). Therefore, using Corollary 8.3(i) and the first inequality in Theorem 8.7(i), we conclude that, increasing the value of the constant D¯ A if necessary, ∇xi V ∞ ≤

D¯ A . n

Corollary 8.9. For every t, r, s, τ ∈ [0, T ], b, b¯ ∈ Td and μ, ν ∈ P(Td ), we have   ¯ − Xst [μ](b)| ≤ D¯ A |τ − s| + |r − t| + |b¯ − b|Td + W2 (ν, μ) . |Xτr [ν](b) Proof. Let t, r, s, τ ∈ [0, T ], q, q¯ ∈ Td and μ, ν ∈ P(Td ). Let γ ∈ 0 (μ, ν). Choose sequences {(xin , yin )}n in the support of γ such that (8.35) holds and n n 1 n n

W 2 μx , μy = |xi − yin |2Td . n

(8.26)

i=1

It follows from Theorem 8.7 and (8.26) that   n ¯ − Xst [μxn ](b)|Td ≤ D¯ A |τ − s| + |r − t| + |b¯ − b|Td + W2 (μx¯ n , μxn ) . |Xτr [μx¯ ](b) Since, by Lemma 4.4, X is continuous, it remains to let n tend to ∞ and use (8.35).

2

8.3. Regularity properties of the master map Let   B = [0, T ] × [0, T ] × Td × (yi , μy ) : i ∈ {1, · · · , n} . Let f : B → R be a continuous function such that if s, t ∈ [0, T ],

q ∈ Td ,

i, j ∈ {1, · · · , n},

x, y ∈ (Td )n

(8.27)

then 

1 |f (t, s, q, (yj , μy )) − f (t, s, q, (xi , μx ))| ≤ CA |xi − yj |Td + W2 μx , μy + . n For z ∈ Td and μ ∈ P(Td ) we define  

 g(t, s, q, z, μ) = inf f (t, s, q, (yj , μy )) + CA |z − yj |Td + W2 μ, μy ,

(8.28)

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where the infimum is performed over the set of (j, y) such that j ∈ {1, · · · , n}, y ∈ (Td )n .

(8.29)

Lemma 8.10. Suppose (8.28) holds whenever (8.27) holds. Suppose that for any x ∈ (Td )n , f (·, · ,·, (xi , μx )) is CA -Lipschitz. Then: √ (i) g is 3CA -Lipschitz. (ii) On B we have |g − f | ≤ CA /n. √ Proof. (i) √ For every t, s ∈ [0, T ], q ∈ Td , g(t, s, q, ·, ·) is 2CA -Lipschitz since it is the ind n fimum of 2CA -Lipschitz functions. This, together √ with the fact that, for every x ∈ (T ) , f (·, ·, ·, (xi , μx )) is CA -Lipschitz, gives that g is 3CA -Lipschitz. (ii) It follows from the definition of g that, for every t, s ∈ [0, T ], q ∈ Td , x ∈ (Td )n , i = 1, . . . , n, g(t, s, q, xi , μx ) ≤ f (t, s, q, (xi , μx )). Moreover, by (8.28),  

 g(t, s, q, xi , μx ) = inf f (t, s, q, (yj , μy )) + CA |xi − yj |Td + W2 μx , μy  

 ≥ inf f (t, s, q, (xi , μx )) + CA |xi − yj |Td + W2 μx , μy 

1 CA − CA |xi − yj | + W2 μx , μy + = f (t, s, q, (xi , μx )) − . n n

2

We now set for t, s ∈ [0, T ], q ∈ Td , x ∈ (Td )n , j = 1, . . . , n, ζ n (t, s, q, (xj , μx )) = n∇xj S(t, s, q, x).

(8.30)

The map ζ n is well defined on B. In particular it is periodic in q and if x ∈ Rd , x¯ ∈ Rd are such that |x = x¯ |Td = 0 then ζ n (t, s, q, (xj , μx )) = ζ n (t, s, q, (x¯j , μx¯ )). Corollary 8.11. For each natural number n, ζ n admits an extension χ n : [0, T ] × [0, T ] × Td × Td × P(Td ) → Rd×d such that, increasing the value of CA , we have: (i) χ n is CA -Lipschitz. (ii) On B we have |χ n − ζ n | ≤ CA /n. Proof. We first check the Lipschitz property of ζ n (·, ·, ·, (xj , μx )) for j ∈ {1, · · · , n} and x ∈ (Td )n . We differentiate with respect to xi the expressions in Lemma 3.16(iv) to obtain ∇xi ∂s st [μx ] = −∇q xi st [μx ]Vts [μx ] − ∇q st [μx ]∇xi Vts [μx ].

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We use the bound on |∇qxi | provided by Corollary 8.3, the bound on |V| provided by Lemma 3.15, the bound on |∇xi Vst [μx ](q)| provided by Remark 8.8, and the fact that ∇q ∞ ≤ A to conclude that ζ n (t, ·, q, xj , μx ) is C˜ A -Lipschitz for some constant C˜ A . Corollary 8.3(i) gives that ζ n (·, s, q, xj , μx ) is CA -Lipschitz, while (8.6) ensures that ζ n (t, s, ·, xj , μx ) is CA -Lipschitz. Corollaries 8.2 and 8.3 imply that ζ n is continuous. It remains to show that ζ n satisfies (8.28) whenever (8.27) holds, so that we may apply Lemma 8.10 to each component of ζ n to conclude the proof. Let t ∈ [0, T ], s ∈ (0, T ), and q ∈ Td . Let x, y ∈ (Td )n and 1 ≤ i < j ≤ n. Since S(t, s, q, x) is invariant under the permutation of the x1 , · · · , xn , and ∇xj S(t, s, q, x) is periodic in the x variables, rearranging and translating the points, we can assume that



|xk − yk |2 ≤ W22 μx , μy

k=i,j

and |xj − yi | = |xj − yi |Td ,

|xi − yj | = |xi − yj |Td .

Moreover, using again the invariance under permutations, we have ∇xj S(t, s, q, y) = ∇x1 S(t, s, q, yj , yi , y1 , · · · , yi−1 , yi+1 , · · · , yj −1 , yj +1 , · · · , yn ), (8.31) and ∇xi S(t, s, q, x) = ∇x1 S(t, s, q, xi , xj , x1 , · · · , xi−1 , xi+1 , · · · , xj −1 , xj +1 , · · · , xn ). (8.32) We combine (8.31) and (8.32) to obtain |∇xj S(t, s, q, y) − ∇xi S(t, s, q, x)| ≤ ∇x1 ,x1 S∞ |yj − xi | + ∇x1 ,x2 S∞ |yi − xj | +

i−1

∇x1 ,xk+2 S∞ |yk − xk |

k=1

+

j −1

∇x1 ,xk+1 S∞ |yk − xk | +

k=i+1

n

∇x1 ,xk S∞ |yk − xk |.

k=j +1

Therefore, by Corollary 8.3, |∇xj S(t, s, q, y) − ∇xi S(t, s, q, x)| ≤

CA CA CA |yk − xk | |yj − xi | + 2 |yi − xj | + 2 n n n k=i,j

≤

√ " CA CA n CA |yk − xk |2 . |yj − xi |Td + 2 |yi − xj |Td + n n n2 k=i,j

´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛

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≤

√

 d CA  |yj − xi |Td + + W2 μ x , μ y , n 2n

where we used the fact that the diameter of Td is n|∇xj S(t, s, q, y) − ∇xi S(t, s, q, x)| ≤

√

√

(8.33)

d/2. Consequently,



1 dCA |yj − xi |Td + W2 μx , μy + , n

which proves that each component of ζ n satisfies (8.28).

2

Theorem 8.12. The following hold: (i) For s ∈ [0, T ], s is differentiable on [0, T ] × Td × P(Td ). (ii) Increasing the value of CA , there is a CA -Lipschitz map ∇¯ μ  : [0, T ] × [0, T ] × Td × Td × P(Td ) → Rd×d such that for any s, r, t ∈ [0, T ], q, ¯ q ∈ Td , μ, ν ∈ P(Td ), and any γ ∈ 0 (μ, ν)   r ¯ − st [μ](q) − ∂t st [μ](q)(r − t) − ∇q st [μ](q)(q¯ − q) s [ν](q)    ∇¯ μ st [μ](q, z)(y − z)γ (dz, dy) − Td ×Td



 ≤ CA (r − t)2 + |q¯ − q|2Td + W22 μ, ν .

(8.34)

Proof. Let μ, ν ∈ P(Td ), r, s, t ∈ [0, T ] and q, ¯ q ∈ Td . Choose a sequence 1 δ(xin ,yin ) n n

γn =

i=1

such that for each i ∈ {1, · · · , n}, (xin , yin ) belongs to the support of γ and {γ n }n converges narrowly to γ in P(Td × Td ). Note that since each (xin , yin ) belongs to the support of γ , {(xin , yin )}ni=1 is | · |Td -monotone (cf. [4]) and so  n  n γ n ∈ 0 μx , μy . Furthermore, n

n

lim W2 μ, μx = lim W2 ν, μy = 0.

n→∞

n→∞

(8.35)

Let the map ζ n be defined as in (8.30) and let χ n be as in Corollary 8.11. By Corollary 8.4

´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛

6625

 n n  r yn ¯ − st [μx ](q) − ∂t st [μx ](q)(r − t) s [μ ](q)   n n  ζ n (t, s, q, z, μx )(y − z)γ n (dz, dy) − ∇q st [μx ](q)(q¯ − q) − Td ×Td

 n n

≤ CA (r − t)2 + |q¯ − q|2 + W22 μx , μy . 

Thus,  n n  r yn ¯ − st [μx ](q) − ∂t st [μx ](q)(r − t) s [μ ](q)   n  t xn χ n (t, q, z, μx )(y − z)γ n (dz, dy) − ∇q s [μ ](q) · (q¯ − q) − Td ×Td

 n n

≤ CA (s − t)2 + |q¯ − q|2 + W22 μx , μy    n n

  ξ n (t, q, z, μx ) − ζ n (t, q, z, μx ) (y − z)γ n (dz, dy). + 

Td ×Td

 C  n n n

n

A ≤ CA (r − t)2 + |q¯ − q|2 + W22 μx , μy + W 2 μx , μ y . n

(8.36)

By Theorem 8.7(ii), {χ n }n is uniformly bounded, and by Corollary 8.11(i) the sequence is CA -Lipschitz. Thus the Arzela–Ascoli lemma provides us with a subsequence {χ nm }m which converges uniformly to a CA -Lipschitz map ∇¯ μ  : [0, T ] × [0, T ] × Td × Td × P(Td ) → Rd×d . By Lemma 4.2,  and ∂t  are continuous while, by Lemma 4.3, ∇q  is continuous. Thus, replacing n by nm in (8.36) and then letting m tend to ∞ we obtain that (ii) is satisfied. 2 Using Corollaries 8.5 and 8.6 and arguing similarly as we did to obtain Corollary 8.11 and then Theorem 8.12, we can derive the following theorem. We leave the details to the readers. Theorem 8.13. The following hold: (i) For s ∈ [0, T ], ∂t s is differentiable on (0, T ) × Td × P(Td ). (ii) Increasing the value of CA , there is a CA -Lipschitz map ξ : [0, T ] × [0, T ] × Td × Td × P(Td ) → Rd×d such that for any s, r, t ∈ (0, T ), q, ¯ q ∈ Td , μ, ν ∈ P(Td ), and any γ ∈ 0 (μ, ν) we have   ¯ − ∂t st [μ](q) − ∂tt st [μ](q)(r − t) − ∇q ∂t st [μ](q)(q¯ − q) ∂t sr [ν](q)    ξst [μ](q, z)(y − z)γ (dz, dy) − Td ×Td

6626

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 ≤ CA (r − t)2 + |q¯ − q|2 + W22 μ, ν .

(8.37)

8.4. Regularity properties of the inverse master map In this subsection, we use the notation of Subsection 8.2. Theorem 8.14. The following hold: (i) X is continuously differentiable on [0, T ] × [0, T ] × Td × P(Td ). (ii) The maps ∂t X, ∇b X are continuous on [0, T ] × [0, T ] × Td × P(Td ) and the map

∇¯ μ Xst [μ](b, z) := −∇b Xst [μ](b) ∇¯ μ st [μ] Xst [μ](b), z is continuous on [0, T ] × [0, T ] × Td × Td × P(Td ). ¯ b ∈ Td , μ, ν ∈ (iii) Increasing suitably the value of D¯ A , we obtain that for any r, t ∈ [0, T ], b, d P(T ), and any γ ∈ 0 (μ, ν), we have   r ¯ − Xst [μ](b) − ∂t Xst [μ](b)(r − t) − ∇b Xst [μ](b)(b¯ − b) Xs [ν](b)    − ∇¯ μ Xst [μ](b, z)(y − z)γ (dz, dy) Td ×Td



 ≤ D¯ A (r − t)2 + |b¯ − b|2 + W22 μ, ν .

(8.38)

Proof. (i) Part (i) will follow once we show (ii) and (iii). (ii) By Lemma 4.4, X, ∂t X and ∇b X are continuous. Since, by Theorem 8.12(ii), ∇¯ μ  is continuous, we conclude the proof of (ii). ¯ b ∈ Td . Set (iii) Let μ, ν ∈ P(Td ), r, t ∈ [0, T ] and b, ¯ i.e. b = st [μ](q), b¯ = sr [ν](q). q = Xst [μ](b), q¯ = Xsr [ν](b), ¯ Recall that, by (3.17) and (3.18), we have

−1 ∇b Xst [μ](b) = ∇q st [μ] ◦ Xst [μ](b) =: E, ∂t Xst [μ](b) = −∇b Xst [μ](b)∂t st [μ] ◦ Xst [μ](b) = −E∂t st [μ] ◦ Xst [μ](b), and, by (3.15), |E| ≤ cd . Therefore   r X [ν](b) ¯ − Xst [μ](b) − ∂t Xst [μ](b)(r − t) + ∇b Xst [μ](b)(b¯ − b)  s    ∇¯ μ Xst [μ](b, z)(y − z)γ (dz, dy) − Td ×Td

  

¯ − st [μ](q) = E ∇q st [μ](q)(q¯ − q) + ∂t st [μ](q)(r − t) − sr [ν](q)

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6627

 t ∇¯ μ σs [μ](q, z)(y − z)γ (dz, dy) 

 + Td ×Td



 ¯ − Xsr [μ](b)|2 + W22 μ, ν . ≤ cd CA (r − t)2 + |Xst [ν](b) It remains to use Corollary 8.9 to find an appropriately large constant D¯ A so that (8.38) is satisfied. 2 8.5. First order expansion of V Recall that, by Definition 3.14, for t, s ∈ [0, T ], q ∈ Td and μ ∈ P(Td ), Vst [μ] := ∂t st [μ] ◦ Xst [μ]. We can differentiate the above expression by Lemma 3.13(iii) to obtain: ∂t Vst [μ] = ∂tt st [μ] ◦ Xst [μ] + ∇q ∂t st [μ] ◦ Xst [μ] ∂t Xst [μ],

(8.39)

∇b Vst [μ] = ∇q ∂t st [μ] ◦ Xst [μ] ∇b Xst [μ].

(8.40)

Set

∇¯ μ Vst [μ](b, z) = ξst [μ] Xst [μ](b), z + ∇q ∂t st [μ] ◦ Xst [μ](b) ∇¯ μ Xst [μ](b, z). Corollary 8.15. The following hold: (i) ∂t V and ∇b V are continuous on [0, T ] × [0, T ] × Td × P(Td ). (ii) ∇¯ μ V is continuous on [0, T ] × [0, T ] × Td × Td × P(Td ). ¯ b ∈ Td , μ, ν ∈ P(Td ), (iii) Increasing the value of D¯ A , we obtain that for any s, r, t ∈ (0, T ), b, and any γ ∈ 0 (μ, ν), we have   r ¯ − Vst [μ](b) − ∂t Vst [μ](b)(r − t) − ∇b Vst [μ](b)(b¯ − b) Vs [ν](b)    − ∇¯ μ Vst [μ](b, z)(y − z)γ (dz, dy) Td ×Td



 ≤ D¯ A (r − t)2 + |b¯ − b|2 + W22 μ, ν .

(8.41)

Proof. (i) By Lemma 4.4, X, ∂t X and ∇b X are continuous. By Lemma 4.3(ii), ∇q ∂t  is continuous, while Lemma 4.2(ii) ensures that ∂tt  is continuous. Using representation formulas (8.39) and (8.40) we thus conclude the proof of (i). (ii) Since X, ∇q ∂t , ξ and ∇¯ μ X are all continuous, we obtain that ∇¯ μ V is continuous. (iii) We combine Theorems 8.13 and 8.14, to obtain

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  r ¯ − Vst [μ](b) − ∂t Vst [μ](b)(r − t) − ∇b Vst [μ](b)(b¯ − b) Vs [ν](b)    ∇¯ μ Vst [μ](b, z)(y − z)γ (dz, dy) − Td ×Td



 ¯ − Xst [μ](b)|2 + W22 μ, ν ≤ CA (r − t)2 + |Xsr [μ](b) 

 + D¯ A ∇q ∂t ∞ (r − t)2 + |b¯ − b|2 + W22 μ, ν 

 ≤ D¯ A (r − t)2 + |b¯ − b|2 + W22 μ, ν , for some appropriately large constant, still denoted by D¯ A . Above we used Corollary 8.9 and the fact that s [μ] ∈ CA to obtain the last inequality. 2 8.6. Smoothness properties of the velocity |Vs |2 We set V¯ = ∇¯ μ VV. Corollary 8.16. The function |Vs |2 is continuously differentiable on (0, T ) × Td × P(Td ) for every s ∈ (0, T ). More precisely, increasing the value of D¯ A as necessary, the following hold: (i) ∂t V · V, ∇q VV and V¯ are continuous. ¯ b ∈ Td , μ, ν ∈ P(Td ), and any γ ∈ 0 (μ, ν) we have (ii) For any s, r, t ∈ (0, T ), b,  |V r [ν](b)| ¯ 2 |Vst [μ](b)|2  s − − (∂t V · V)ts [μ](b)(r − t)  2 2    V¯ st [μ](b, z) · (y − z)γ (dz, dy) − (∇q VV)ts [μ](b) · (b¯ − b) − Td ×Td



 ≤ D¯ A (r − t)2 + |b¯ − b|2 + W22 μ, ν .

(8.42)

Proof. The continuity of ∂t V · V, ∇q VV and V¯ follows from the continuity of V, ∂t V, ∇q V and ∇¯ μ V. Part (ii) is a direct consequence of Corollary 8.15. 2 9. Strong solutions to the master equation Throughout this section we assume that T > 0, A > 0 satisfy (3.7). As in Section 6 we assume that F, F, u∗ and U∗ are given through functions φ, U 0 , and U 1 satisfying the assumptions imposed in that section. Using the notation of Section 7, we define u : [0, T ] × Td × P(Td ) → R

´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛

6629

by

u(s, q, μ) = u∗ q, s0 [μ]# μ −

s 



 |Vsl [μ](q)|2 + F q, (sl [μ])# μ dl 2

(9.1)

0

for s ∈ [0, T ], q ∈ Td and μ ∈ P(Td ). 9.1. Regularity of u with respect to the μ variable We set  Nst [μ](q) = −



∇φ q − st [μ](y) · ∂t st [μ](y)μ(dy)

Td

and

N¯ st [μ](q, y) = −(∇q st [μ](y))T ∇φ q − st [μ](y) 

− (∇¯ μ st [μ](u, y))T ∇φ q − st [μ](u) μ(du), Td

where (∇q st [μ](y))T and (∇¯ μ st [μ](u, y))T are the transpositions of the matrices ∇q st [μ](y) and ∇¯ μ st [μ](u, y).

Lemma 9.1. The function (t, s, q, μ) → F q, st [μ]# μ is continuously differentiable in the following sense:

(i) N and (t, s, q, μ) → ∇q F q, st [μ]# μ are continuous on [0, T ] × [0, T ] × Td × P(Td ), and N¯ is continuous on [0, T ] × [0, T ] × Td × Td × P(Td ). (ii) Suitably increasing the value of D¯ A we have that for any (s, t, r, q, q, ¯ μ, ν) ∈ [0, T ] × [0, T ] × [0, T ] × Td × Td × P(Td ) × P(Td ) and any γ ∈ 0 (μ, ν), 

 ¯ σ¯ r − F (q, σt ) − (r − t)Nst [μ](q) − ∇q F (q, σt ) · (q¯ − q) F q,    N¯ st [μ](q, y) · (z − y)γ (dy, dz) − Td ×Td

  ≤ D¯ A (r − t)2 + |q¯ − q|2 + W22 (μ, ν) . Here, σt = st [μ]# μ,

σ¯ r = sr [ν]# ν.

(9.2)

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Proof. (i) The continuity of N and (t, s, q, μ) → ∇q F q, st [μ]# μ follows from the continuity of ∇φ, V and . Since in addition ∇q  and ∇¯ μ  are continuous, we conclude that N¯ is continuous. (ii) We have 





φ q¯ − sr [ν](z) − φ q − st [μ](y) γ (dy, dz). F q, ¯ σ¯ r − F (q, σt ) = Td ×Td

Since ∇ 2 φ∞ ≤ κ, we obtain   # 



 φ q¯ − sr [ν](z) − φ q − st [μ](y)   Td ×Td

 $ 

r

 t t − ∇φ q − s [μ](y) · (q¯ − q) − s [ν](z) − s [μ](y) γ (dy, dz)     κ  2 ≤ q¯ − q − sr [ν](z) + st [μ](y) γ (dy, dz) 2 Td ×Td



≤ C˜ A

  (r − t)2 + |q¯ − q|2 + |z − y|2 + W22 (μ, ν) γ (dy, dz)

Td ×Td

  = C˜ A (r − t)2 + |q¯ − q|2 + 2W22 (μ, ν) ,

(9.3)

for some independent constant C˜ A . We notice that 

∇φ q − st [μ](y) · (q¯ − q)γ (dy, dz) = ∇q F (q, σt ) · (q¯ − q). Td ×Td

Using Theorem 8.12(ii) we now have   # 

 t ∇φ q − s [μ](y) · sr [ν](z) − st [μ](y) − ∂t st [μ](y)(r − t)   Td ×Td



− ∇q st [μ](y)(z − y) −  ≤ κCA

   γ (dy, dz) 

$ ∇¯ μ st [μ](y, w)(u − w)γ (dw, du)

Td ×Td

  (r − t)2 + |z − y|2 + W22 (μ, ν) γ (dy, dz)

Td ×Td

  = κCA (r − t)2 + 2W22 (μ, ν) . It remains to combine (9.3) and (9.4) and notice that

(9.4)

´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛





∇φ q − st [μ](y) · ∂t st [μ](y)μ(dy)

−Nst [μ](q) = Td

6631





∇φ q − st [μ](y) · ∂t st [μ](y)γ (dy, dz),

= Td ×Td

and 

#

t ∇φ q − s [μ](y) · ∇q st [μ](y)(z − y)

Td ×Td

 +

$ t ¯ ∇μ s [μ](y, w)(u − w)γ (dw, du) γ (dy, dz)

Td ×Td



=−

N¯ st [μ](q, y) · (z − y)γ (dy, dz).

Td ×Td

To obtain the last equality we first changed the order of integration and then renamed the variables. 2 Denote

¯ s [μ](q, y) = −(∇q s0 [μ](y))T ∇U 1 q − s0 [μ](y) M 

− (∇¯ μ s0 [μ](u, y))T ∇U 1 q − s0 [μ](u) μ(du). Td

Repeating the same proof as this of Lemma 9.1 we also obtain the following result.

Lemma 9.2. The function (s, q, μ) → u∗ q, s0 [μ]# μ is continuously differentiable in the following sense:

¯ (i) The function (s, q, μ) → ∇q u∗ q, s0 [μ]# μ is continuous on [0, T ] × Td × P(Td ), and M d d d is continuous on [0, T ] × T × T × P(T ). (ii) Suitably increasing the value of D¯ A we have that for any (s, q, q, ¯ μ, ν) ∈ [0, T ] × Td × Td × P(Td ) × P(Td ) and γ ∈ 0 (μ, ν) 

 ¯ σ¯ 0 − u∗ (q, σ0 ) − ∇q u∗ (q, σ0 ) · (q¯ − q) u∗ q,  − Td ×Td

 ¯ 0s [μ](q, y) · (z − y)γ (dy, dz) M

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  ≤ D¯ A |q¯ − q|2 + W22 (μ, ν) ,

(9.5)

where σt , σ¯ r are as in Lemma 9.1. We now set

ϒs [μ](q, y) =

s 

 ¯ 0s [μ](q, y). V¯ st [μ](q, y) + N¯ st [μ](q, y) dt + M

0

The following corollary is a direct consequence of Corollary 8.16 and Lemmas 9.1 and 9.2. Corollary 9.3. The following hold: (i) ϒ is continuous on [0, T ] × Td × Td × P(Td ). (ii) Suitably increasing the value of D¯ A we have that for any (s, q, μ, ν) ∈ [0, T ] × Td × P(Td ) × P(Td ) and γ ∈ 0 (μ, ν)      ϒs [μ](q, y) · (z − y)γ (dy, dz) ≤ D¯ A W22 (μ, ν). u(s, q, ν) − u(s, q, μ) − Td ×Td

¯ N¯ and M ¯ are continuous, we obtain that ϒ is continuous. Proof. (i) Since V, (ii) We combine Corollary 8.16 and Lemmas 9.1 and 9.2 to obtain (ii). 2 Remark 9.4. (i) We combine Remark 2.6 and Corollary 9.3 to obtain that the gradient of u(s, q, ·) at μ is the orthogonal projection of ϒ(s, q, y, μ) onto the tangent space Tμ P(Td ). Since the combination of Lemma 6.6(ii) and Lemma 7.1(i) gives that vs belongs to the tangent space and vs = ∇q u(s, ·, μ), we obtain 

 ϒs [μ](q, y) · vs (y)μ(dy) =

Td

∇μ u(s, q, μ)(y) · ∇q u(s, ·, μ)μ(dy). Td

(ii) Since ϒ is continuous on a compact set, it is bounded and so, ∇μ u is bounded. Using Corollary 9.3 we thus obtain that there exists a constant κ1 which is independent of s ∈ [0, T ] and q ∈ Td , such that u(s, q, ·) is κ1 -Lipschitz. 9.2. Existence of a strong solution to the master equation Theorem 9.5. The function u defined in (9.1) is Lipschitz on [0, T ] × Td × P(Td ), differentiable with respect to each of its variables, and u(0, ·, ·) = u∗ . Furthermore, u satisfies the following:

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6633

(i) For any s ∈ [0, T ] and any μ ∈ P(Td ), there exists σ¯ ∈ AC2 (0, T ; P(Td )) such that σ¯ s = μ and

in D (0, T ) × Td .

∂t σ¯ t + ∇ · (σ¯ t ∇q u(t, q, σ¯ t )) = 0

(ii) The gradient ∇q u and the derivative ∂t u are continuous on (0, T ) × Td × P(Td ) and, for every (s, q, μ) ∈ (0, T ) × Td × P(Td ),  ∂t u(s, q, μ) +

∇μ u(s, q, μ)(y) · ∇q u(s, y, μ)μ(dy) +

|∇q u(s, q, μ)|2 − F (q, μ) = 0. 2

Td

Proof. We know from Lemma 7.1 and Corollary 9.3 that ∇q u and ∇μ u exist on (0, T ) × Td × P(Td ), and Lemma 4.2(ii) guarantees that ∇q u is continuous. Moreover, Lemma 3.15 and Remark 9.4 allow us to conclude that the functions u(t, ·, ·) are Lipschitz continuous with a Lipschitz constant which is independent of t ∈ [0, T ]. It thus suffices to show that the derivative ∂t u exists for every (s, q, μ) ∈ (0, T ) × Td × P(Td ) and that (ii) is satisfied to conclude that u is Lipschitz. We will use the following notation: v¯ t (q) = Vst [μ],

σ¯ t = st [μ]# μ,

and     σˆ t = Id + (t − s)vs σ¯ s = ss [μ] + (t − s)∂t st=s [μ] μ. #

#

(i) By Lemma 7.1, ∇q u(t, q, σ¯ t ) = v¯ t (q) and so, since the function U used in Corollary 7.2 satisfies ∇q u(t, q, σ¯ t ) = ∇q U (t, q), we use (ii) of the same corollary to conclude that (i) holds for σ¯ t . (ii) If s + h ∈ [0, T ] then 

 gh := ss+h [μ] × Id + h¯vs μ ∈ (σ¯ s+h , σˆ s+h ), #

and thus, W22 (σ¯ s+h , σˆ s+h ) ≤

  2  s+h  s [μ](q) − ss [μ](q) − h∂t ss [μ](q) d μ(dq) Td

≤

h4 ||∂tt ||2∞ . 4

T

(9.6)

By Lemma 3.15, v¯ ∈ C 1 (Td )d and so, by Lemma 6.6(ii), v¯ is the gradient of a function which belongs to C 2 (Td ). Thus, for |h| small enough, Id + h¯vs is the gradient of a convex function. Consequently (see e.g. [4])

γh := Id × (Id + h¯vs ) # μ ∈ 0 (μ, σˆ s+h ),

(9.7)

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6634

and so  W22 (σˆ s+h , μ) =

|a − b|2 γh (da, db) = h2 ¯vs 2μ .

(9.8)

Td ×Td

Furthermore, using (9.7) in Corollary 9.3, we have       

  u s + h, q, σˆ s+h − u(s + h, q, μ) − h ϒs+h [μ](q, y) · v¯ s (y)μ(dy)     Td ≤ D¯ A W22 (σˆ s+h , μ). This, together with the fact that ϒ is continuous on a compact set and thus admits a modulus of continuity ω, yields       

  u s + h, q, σˆ s+h − u(s + h, q, μ) − h ϒs [μ](q, y) · v¯ s (y)μ(dy)     Td ≤ |h|ω(|h|)¯vs μ + D¯ A W22 (σˆ s+h , μ).

(9.9)

By Remark 9.4, for any t ∈ [0, T ], u(t, q, ·) is κ1 -Lipschitz and so, thanks to (9.6), 



 κ1 h2  ∂tt ∞ . u s + h, q, σˆ s+h − u s + h, q, σ¯ s+h  ≤ 2

(9.10)

By Lemma 7.1     



u s + h, q, σ¯ s+h − u(s, q, μ + h 1 |∇q u(s, q, μ)|2 + F (q, μ)  = o(h).   2

(9.11)

Writing

u(s + h, q, μ) − u(s, q, μ) = u(s + h, q, μ) − u s + h, q, σˆ s+h



+ u s + h, q, σˆ s+h − u s + h, q, σ¯ s+h

+ u s + h, q, σ¯ s+h − u(s, q, μ), we obtain    u(s + h, q, μ) − u(s, q, μ) + h ϒs [μ](q, y) · vs (y)μ(dy) Td

1   + h |∇q u(s, q, μ)|2 + F (q, μ)  2

(9.12)

´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛



 ≤ u(s + h, q, μ) − u s + h, q, σˆ s+h + h



6635

  ϒs [μ](q, y) · v¯ s (y)μ(dy)

Td





 + u s + h, q, σˆ s+h − u s + h, q, σ¯ s+h   1 

  + u s + h, q, σ¯ s+h − u(s, q, μ) + h |∇q u(s, q, μ)|2 + F (q, μ) . 2

(9.13)

We combine (9.8), (9.9), (9.10) and (9.13) to conclude that if we set 

1 ϒs [μ](q, y) · v¯ s (y)μ(dy) − |∇q u(s, q, μ)|2 − F (q, μ) 2

u¯ := − Td

then |u(s + h, q, μ) − u(s, q, μ) − hu| ¯ = o(h). This proves that u(·, q, μ) is differentiable at s, ∂t u(s, q, μ) = u, ¯ and ∂t u is continuous on (0, T ) × Td × P(Td ). In other words,  ϒs [μ](q, y) · v¯ s (y)μ(dy) +

∂t u(s, q, μ) +

|∇q u(s, q, μ)|2 + F (q, μ) = 0. 2

Td

We now use Remark 9.4(i) to complete the proof.

2

Definition 9.6. We say that a continuous function v : [0, T ] × Td × P(Td ) → R is a strong solution to (1.1) if v(0, ·, ·) = u∗ , v is differentiable with respect to each of its variables on (0, T ) × Td × P(Td ), ∂t v, ∇q v are bounded and continuous on (0, T ) × Td × P(Td ), there exists a bounded and continuous map ϒ : (0, T ) × Td × Td × P(Td ) → Rd satisfying for every (t, q, μ, ν) ∈ (0, T ) × Td × P(Td ) × P(Td ) sup

γ ∈ 0 (μ,ν)

   q, ν)−v(t, q, μ)− v(t,

  ϒs [μ](q, y)·(z−y)γ (dy, dz) = o (W2 (μ, ν)) , (9.14)

Td ×Td

and we have  ∂t v(t, q, μ) +

∇μ v(t, q, μ)(y) · ∇q v(t, y, μ)μ(dy) +

|∇q v(t, q, μ)|2 − F (q, μ) = 0 2

Td

for every (t, q, μ) ∈ (0, T ) × Td × P(Td ). A direct consequence of Theorem 9.5 is the following corollary. Corollary 9.7. The function u defined in (9.1) is a strong solution to (1.1).

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We finish this section with a chain rule for functions regular enough to be strong solutions to (1.1). In the rest of this section T is any positive number. d Lemma 9.8. Let T > 0 and let v : (0, T ) × Td × P(T ) →d R have the regularity properties

1,2 required for a strong solution to (1.1). Let Q ∈ W 0, T ; T , σ ∈ AC2 0, T ; P(Td ) . Let s ∈ ˙ s exists and there exists a velocity of minimal norm vs for σ . (Recall that (0, T ) be such that Q by Theorem 8.3.1 and Proposition 8.4.5 of [4], vt exists for a.e. t and vt ∈ Tσt P(Td ).) Then the function t → v(t, St , σt ) is differentiable at t = s and

d ˙s v(t, St , σt )|t=s = ∂t v(s, Qs , σs ) + ∇q v(s, Qs , σs ) · Q dt  + ∇μ v(s, Qs , σs )(y) · vs (y)σs (dy).

(9.15)

Td

Proof. The proof is similar to the proof of Theorem 9.5. We define   σˆ t := Id + (t − s)vs σs . #

Then

γh := Id × (Id + h¯vs ) # σs ∈ 0 (σs , σˆ s+h ), and, by (9.8), W2 (σˆ s+h , σs ) = |h|¯vs σs .

(9.16)

Moreover, by Proposition 8.4.6 of [4], W2 (σs+h , σˆ s+h ) = o(h).

(9.17)

We have v(s + h, Qs+h , σs+h ) − v(s, Qs , σs ) = v(s + h, Qs+h , σs+h ) − v(s, Qs+h , σs+h ) + v(s, Qs+h , σs+h ) − v(s, Qs , σs+h ) + v(s, Qs , σs+h ) − v(s, Qs , σs ).

(9.18)

By the mean value theorem there is τ ∈ [s, s + h] such that v(s + h, Qs+h , σs+h ) − v(s, Qs+h , σs+h ) = h∂t v(τ, Qs+h , σs+h ). Therefore, by the continuity of ∂t v,    v(s + h, Qs+h , σs+h ) − v(s, Qs+h , σs+h )  o(h)  − ∂t v(s, Qs , σs ) = .  h |h|

(9.19)

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6637

Using the mean value theorem again, there is z in the line segment connecting Qs and Qs+h such that v(s, Qs+h , σs+h ) − v(s, Qs , σs+h ) = ∇q v(s, z, σs+h ) · (Qs+h − Qs ). Therefore, by the continuity of ∇q v,    v(s, Qs+h , σs+h ) − v(s, Qs , σs+h )  o(h)  ˙ − ∇q v(s, Qs , σs ) · Qs  = .  h |h|

(9.20)

Now v(s, Qs , σs+h ) − v(s, Qs , σs ) = v(s, Qs , σs+h ) − v(s, Qs , σˆ s+h ) + v(s, Qs , σˆ s+h ) − v(s, Qs , σs ).

(9.21)

Since it is easy to see that v is Lipschitz, there is a constant L > 0, such that    v(s, Qs , σs+h ) − v(s, Qs , σˆ s+h )  LW 2 (σs+h , σˆ s+h ) o(h)  ≤ = ,   h |h| |h|

(9.22)

where we used (9.17). Finally, if ϒ is a function from (9.14) for v, we obtain   v(s, Q , σˆ ) − v(t, q, σ )  s s+h s   − ϒs [σs ](Qs , y) · vs (y)σs (dy)  h Td

 v(s, Q , σˆ ) − v(t, q, σ ) 1  s s+h s  − = h h =

o



Td ×Td



W22 (σs , σˆ s+h ) |h|

  ϒs [σs ](Qs , y) · (z − y)γh (dy, dz)

=

o(|h|¯vs σs ) o(h) = , |h| |h|

(9.23)

where we used (9.16). Therefore, combining (9.18), (9.19), (9.20), (9.21), (9.22) and (9.23), we obtain d v(t, St , σt )|t=s = ∂t v(s, Qs , σs ) + ∇q v(s, Qs , σs ) · Q˙ s dt  + ϒs [σs ](Qs , y) · vs (y)σs (dy). Td

It remains to notice that vs ∈ Tσs P(Td ), and thus 

 ϒs [σs ](Qs , y) · vs (y)σs (dy) =

Td

∇μ v(s, Qs , σs )(y) · vs (y)σs (dy). Td

2

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6638

9.3. Connection with MFG equations and existence of a Nash equilibrium Our study does not establish whether strong solutions to (1.1) are unique and thus we cannot exclude the possibility that there may be another strong solution to (1.1) not given by the representation formula (9.1). For this reason, in this subsection, without appealing to that representation formula, we explain how any strong solution to (1.1) can be used to obtain a solution to the First Order Mean Field Games equations (1.2) and obtain the existence of an analogue of a Nash equilibrium for a game with a continuum of players. We begin with a lemma that provides a rigorous connection between strong solutions to the master equation (1.1) and the First Order Mean Field Games equations (1.2). Lemma 9.9. Let T > 0, let u be a strong solution to (1.1) (see Definition 9.6), and let μ ∈ P(Td ). Then:

(i) There exist σ¯ ∈ AC2 0, T ; P(Td ) such that 

∂t σ¯ t + ∇ · ∇q u(t, q, σ¯ t )σ¯ t = 0 σ¯ T = μ.



in D (0, T ) × Td ,

(9.24)

The solution σ¯ is given by σ¯ t = S(t, ·)# μ for t ∈ [0, T ] and if μ is non-atomic, so is σ¯ t for t ∈ [0, T ]. In particular if μ has a density with respect to the Lebesgue measure, so does σ¯ t for t ∈ [0, T ]. Here, S is the flow uniquely determined by the system of differential equations 

∂t S(t, q) = ∇q u(t, S(t, q), σ¯ t ), S(s, q) = q

q ∈ Td , t ∈ (0, T ) q ∈ Td .

(9.25)



(ii) If U (t, q) := u(t, q, σ¯ t ), then U ∈ C([0, T ] × Td ) ∩ C 1 (0, T ) × Td and the pair (σ¯ , U ) satisfies the system of equations (1.2), in fact U is a classical solution to the HJ equation in this system. Proof. We sketch the proof. Since ∇q u is continuous and bounded, for any σ ∈ AC2 (0, T ,

P(Td )) there exists σ ∗ ∈ AC2 0, T ; P(Td ) such that 

∂t σt∗ + ∇ · ∇q u(t, q, σt )σt∗ = 0 σs∗ = μ.



in D (0, T ) × Td ,

(9.26)

In other words, we have defined a map which to each σ , associates σ ∗ . Using iterations one checks that this map has a fixed point σ¯ ; in other words, σ¯ satisfies (9.24). It is obvious that U (t, q) := u(t, q, σ¯ t ) satisfies U ∈ C([0, T ] × Td ). By Lemma 9.8 and the fact that u is a strong solution to (1.1) we have, for a.e. t ∈ (0, T ) and every q ∈ Td , ∂t U (t, q) =

d u(t, q, σ¯ t ) dt



= ∂t u(t, q, σ¯ t ) +

∇μ u(t, q, σ¯ t )(y) · vt (y)σ¯ t (dy) Td

´ ech / J. Differential Equations 259 (2015) 6573–6643 W. Gangbo, A. Swi˛

=−

|∇q u(t, q, σ¯ t )|2 − F (q, σ¯ t ), 2

6639

(9.27)

since vt = ∇q u(t, q, σ¯ t ) for a.e. t . Noticing that the right hand side of (9.27) is continuous, we conclude that ∂t U (t, q) must exist for every t ∈ (0, T ) and every q ∈ Td and be equal to the right hand side of (9.27). Thus U ∈ C 1 ((0, T ) × Td ) and the pair (σ¯ , U ) solves (1.2). In particular U is a classical solution to the HJ equation ∂t U (t, q) +

|∇q U (t, q)|2 + F (q, σ¯ t ) = 0, 2

U (0, q) = u∗ (0, q, σ¯ 0 ).

(9.28)

Moreover it is a standard that, under our assumptions on u∗ and F , we have ∇qq U ≤ CId for some C > 0. Using these facts it then follows from the theory of HJ equations and ODE theory that (9.25) has a unique solution on (0, T ) and the flow satisfies |S(t, q1 ) − S(t, q2 )|Td ≥ C1 |q1 − q2 |Td for t ∈ [0, T ], q1 , q2 ∈ Td , for some C1 > 0, and thus if μ is non-atomic, so is σˆ t := S(t, ·)# μ for t ∈ [0, T ]. In particular if μ has a density, so does σˆ t := S(t, ·)# μ for t ∈ [0, T ]. Moreover, by results on the continuity equation, σˆ is the unique solution to 

∂t σˆ t + ∇ · ∇q u(t, q, σ¯ t )σˆ t = 0 σˆ T = μ.



in D (0, T ) × Td ,

(9.29)

For the proofs of these statements we refer the reader to [12], Section 4.1, Lemmas 4.11 and 4.13, and Section 4.2, Theorem 4.18 (see also [1]). The uniqueness of solutions of (9.29) thus implies σ¯ t = σˆ t = S(t, ·)# μ, which completes the proof. 2 We assume in the rest of this subsection that T > 0 and u is a strong solution to (1.1). We are now ready to explain what we mean by a Nash equilibrium. Let s ∈ [0, T ] and μ ∈ P(Td ) be a non-atomic measure; for instance, we may assume that μ has a density. Given paths

Q ∈ W 1,2 0, s; Td ,



σ ∈ AC2 0, s; P(Td ) ,

we define the action Js (Q, σ ) =

s 

 1 ˙ 2 |Qτ | − F (Qτ , στ ) dτ 2

0

and the augmented action Jso (Q, σ ) =

s  ˙ 2

 |Qt | − F Qt , σt dt + u∗ (Q0 , σ0 ). 2 0

Suppose we have infinitely many players represented by points on Td , the position of an average player at time s is at q ∈ Td , and μ is the given probability distribution of the players at time s. The players try to choose paths to minimize their utility functions given by the augmented

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actions. It is assumed that each player knows the overall distribution of all the players at each time t, which is represented by the measure σ [S]t , and which is determined by the overall flow of all players. However neither player can change the distribution of the players by his/her own actions alone as it can be changed only by a collective action of the players. This is why the phrase continuum of players is used and games with this kind of structure are also called non-atomic. We are thus looking for a map S(t, q) such that, S(·, q) ∈ W 1,2 (0, s; Td ) for every q ∈ Td , and



Jso S(·, q), σ [S] ≤ Jso Q, σ [S]

for every q ∈ Td , Q ∈ W 1,2 0, s; Td such that Q(s) = q, where σ [S]t = S(t, ·)# μ. The measure σ [S]t gives the distribution of the players at time t determined by the flow S, and the path S(·, q) is then optimal for the player which is located at q at time s for every q ∈ Td . This is what we mean by a Nash equilibrium. Let S, σ¯ be as in Lemma 9.9 (applied with T = s) so that (9.24) and (9.25) are satisfied and σ¯ t has a density with respect to the Lebesgue measure for all t ∈ [0, s]. We recall that if u is the strong solution constructed in Section 9.2 then the pair S(t, q) = st [μ](q),

σ¯ t = S(t, ·)# μ

solves (9.24)–(9.25). We refer to ∇q u(t, q, σ¯ t ) as a closed loop feedback control strategy. We claim that the map S constructed this way gives a Nash equilibrium for the game in the sense described above, and u(t, q, μ) is the payoff function for the player which is at y at time s, i.e., for every y, u(s, q, μ) = Jso (S(·, q), σ¯ ) ≤ Jso (Q, σ¯ ) for every path Q : [0, s] → Td such that Qs = q. We refer the reader to [12] and [22] for more on the concept of a Nash equilibrium for games with large numbers of players.

Lemma 9.10. Let Q ∈ W 1,2 0, s; Td and be differentiable at t ∈ (0, s). Then   d u(t, Qt , σ¯ t ) − Jt (Q, σ¯ ) < 0 dt ˙ t = ∇q u(t, Qt , σ¯ t ), in which case equality holds. unless Q Proof. We remind that v¯ t = ∇q u(t, ·, σ¯ t ) is the velocity of minimal norm for σ¯ for every t . We thus have, by (9.28),     d d u(t, Qt , σ¯ t ) − Jt (Q, σ¯ ) = U (t, Qt ) − Jt (Q, σ¯ ) dt dt ˙ 2 ˙ t · ∇q U (t, Qt ) − |Qt | + F (Qt , σ¯ t ) = ∂t U (t, Qt ) + Q 2

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1 1 ˙ 2 = ∂t U (t, Qt ) + |∇q U (t, Qt )|2 + F (Qt , σ¯ t ) − |Q t − ∇q U (t, Qt )| 2 2 1 = − |Q˙ t − ∇q u(t, Qt , σ¯ t )|2 . 2 This completes the proof.

2



Corollary 9.11 (Existence of a Nash equilibrium). Assume Q ∈ W 1,2 0, s; Td is such that Qs = q. Then u(s, q, μ) = Jso (S(·, q), σ¯ ) < Jso (Q, σ¯ ),

(9.30)

unless Q ≡ S(·, q). ˙ t ≡ ∇q u(t, Qt , σ¯ t ) for a.e. t ∈ [0, s], we have Proof. By Lemma 9.10, unless Q u(s, Qs , μ) − Js (Q, σ¯ ) − u(0, Q0 , σ¯ 0 ) < 0, and we have u(s, S(s, q), μ) − Js (S(·, q), σ¯ ) − u(0, S(0, q), σ¯ 0 ) = 0. We now use that u(s, Qs , μ) = u(s, q, μ) = u(s, S(s, q), μ) and u(0, Q0 , σ¯ 0 ) = u∗ (Q0 , σ¯ 0 ),

u(0, S(0, q), σ¯ 0 ) = u∗ (S(0, q), σ¯ 0 ),

˙ t ≡ ∇q u(t, Qt , σ¯ t ) for a.e. t ∈ to obtain (9.30). Since S(·, q) is the unique solution to (9.25), Q [0, s], implies S(·, q) ≡ Q on [0, s]. 2 Acknowledgments The research of W. Gangbo was supported by NSF grant DMS-1160939. W. Gangbo would like to acknowledge the generous support provided through the Eisenbud Professorship at MSRI during the Fall 2013 program on Optimal Transportation: Geometry and Dynamics, where part of this work was done. W. Gangbo would also like to thank R. Jensen, R. Hynd and K. Trivisa for many extremely fruitful discussions at MSRI. Both authors would like to thank P. Cardialaguet for his input on the first draft of the manuscript. References [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2) (2004) 227–260. [2] L. Ambrosio, J. Feng, On a class of first order Hamilton–Jacobi equations in metric spaces, J. Differential Equations 256 (7) (2014) 219–2245. [3] L. Ambrosio, W. Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures, Comm. Pure Appl. Math. 61 (1) (2008) 18–53.

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