An infinite-dimensional Weak KAM theory via random variables

We develop several aspects of the infinite-dimensional Weak KAM theory using a random variables' approach. We prove that the infinite-dimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we show the existence of invariant minimizing measures and calibrated curves defined on R.


Introduction
In this paper, we study dynamical systems with an infinite number indistinguishable particles on a d-dimensional torus T d . We extend and apply methods from the Weak KAM theory [16,20,18,17,19,21,22,23,24,6,7,5,29,30,31,32] to the infinite-dimensional setting using a random variables' approach. In particular, we prove that the infinitedimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we construct invariant subsets and invariant minimizing measures under the Lagrangian dynamics. Finally, we obtain the existence of calibrated curves defined on R.
Infinite-dimensional systems (infinite systems for short) arise in the study of mechanical systems with a large number of identical particles (e.g. fluids and gases). In these models, the number of particles is infinite and the state of the system is determined by a probability measure or, alternatively, by a random variable. The evolution of the system is characterized by an ODE in the space of probability measures or the space of random variables. A well-known example of such an ODE is the non-linear Vlasov system [12,15,36,26,27,28].
In the seminal papers [26,27], Gangbo and Tudorascu introduced and developed the weak KAM theory for infinite systems. Next, they considered an infinite system of particles on the torus T 1 and modeled it using L 2 ([0, 1]) functions as random variables. They introduced the infinite-dimensional torus and proved a Weak KAM theorem on it. Sub-sequently, in [28], these authors addressed the higher-dimensional case on the torus T d for d > 1. This case is studied using probability measures over T d . They generalize core aspects of the one-dimensional problem to higher dimensions.
In [8,9,10,11], Bessi examined infinite systems in the framework of the Aubry-Mather theory. In particular, in [9], the author studied the Aubry-Mather minimal measure theory in the infinite-dimensional setting for d = 1.
Crucial aspects of the previous results are the following. Firstly, in the one-dimensional case, the existence of optimal trajectories is proved for monotone and square integrable initial configurations of particles. This technique is used to overcome the fact that L 2 ([0, 1]) is not locally compact. Monotonicity yields compactness, and that makes it possible to extend finite-dimensional methods. Unfortunately, this technique does not generalize to higher dimensions.
Secondly, in prior publications, the higher-dimensional case was studied via the probability measures approach. The space of probability measures is a metric space, and it does not have a natural linear structure. This fact creates additional difficulties. A standard solution to define the velocity of a curve is to consider velocity fields of minimal norm [1]. As pointed out in [28], this is not the suitable notion to develop the Weak KAM on the space of probability measures. The appropriate derivatives are the c-minimal velocity fields. However, these depend on the choice of c ∈ R d . Therefore, the definition of the viscosity solution of the cell problem depends on c.
Thirdly, for any dimension d ≥ 1, there exist weakly invariant minimizing measures (or minimizing holonomic measures) [26,28]. Moreover, if d = 1 there exist invariant (or strongly invariant) minimizing measures [9]. The existence of invariant minimizing measures in the case d > 1 was not settled previously.
Finally, the Lagrangians considered in previous publications are mechanical Lagrangians that are the sum of kinetic and potential energy.
In this paper, we contribute to the existing results in several directions. For any dimension d ≥ 1, we address the following points: i. For any c ∈ R d and for generic initial configurations of the particles, we prove the existence of the optimal trajectories for the discounted cost infinite horizon problem.
ii. For any c ∈ R d , we prove that the infinite-dimensional cell problem admits a viscosity solution U and that this solution is a fixed point for Lax-Oleinik semigroup. Moreover, we show the existence of (U, c, L)-calibrated curves defined on R + , where L is the Lagrangian of the system.
iii. We show the existence of invariant minimizing measures and (U, c, L)-calibrated curves defined on R.
In what follows, we present the statements of our main results and give a detailed description of our methods.
In this paper, we assume that the Lagrangian L : L 2 (I; R d ) × L 2 (I; R d ) → R satisfies the conditions i)− viii) given in Subsection 2.4. An important example that satisfies these conditions is the mechanical Lagrangian where W ∈ C 2 (T d ) is an interaction potential.
In Section 3, we consider the discounted-cost infinite-horizon problem.
We fix ε > 0, and for a trajectory x ∈ AC 2 loc ((0, ∞); L 2 (I; R d )), we define the action Since L c is bounded by below, A ε is well defined. Set The Hamiltonian is the Legendre transform of L given by The Hamiltonian H c associated with L c is given by We have that (see [14,34,40,33,9]) V ε is a viscosity solution of Our first result is: ) and x * solves the Euler-Lagrange equation, that is To date, the existence of the minimizers for higher dimensions has been an open problem (see Remark 3.4 in [26]). Our proof is based on the techniques that we developed regarding the existence of minimizers of the optimal control problem in Hilbert spaces [38,33].
In Section 4, we present the proofs of our main results. Firstly, we extract a convergent subsequence out of the family of the functions {εV ε } and {U ε : We prove the following theorem. Additionally, for every differentiability point M ∈ L 2 (I; R d ) of U , there exists a unique trajectory x * ∈ C 1 ([0, ∞); L 2 (I; R d )) whose restriction to the interval [0, T ] is a minimizer of for any T > 0. The infimum is taken over the curves in AC 2 ((0, T ); L 2 (I; R d )). Moreover, x * satisfies the Euler-Lagrange equation Our next key result is: For any c ∈ R d , there exists a closed infinite-dimensional subset Ω of the tangent bundle T L 2 (I; R d ) that is invariant under the Euler-Lagrange flow (2.4).
As corollaries to this Theorem, we get: , that is, (4.4) holds.
where the infimum is taken over all invariant probability measures µ on (Ω, B). Moreover, the infimum is achieved.

Preliminaries and main assumptions
Here, we present background material on mechanical systems with finite or infinite number of identical particles.

Mechanical systems with a finite number of indistinguishable particles
Consider a system of n identical particles on the torus T d . Let l : (R d ) n × (R d ) n → R be the corresponding Lagrangian. Denote by x i (t) ∈ R the position of the particle "i" at time t. Let (x 0 i , v 0 i ) be the corresponding initial position and velocity. From Hamilton's minimal action principle, the system evolves according to the Euler-Lagrange equation: Then, (2.1) has the equivalent formulation in Hamiltonian form where p(t) = (p 1 (t), p 2 (t), · · · p n (t)) ∈ (R d ) n is referred to as the momentum.
A critical issue in classical mechanics is the study of qualitative properties of (2.1) and (2.2). Since the particles move on the torus T d , we assume that l (and, consequently, h) are periodic in the position variable, x. Because the particles are identical, l and h are invariant under permutations, that is, for all points (x i , v i ) ∈ R d × R d , i = 1, 2, · · · , n and all permutations σ ∈ S n , Then, (2.1) and (2.2) can be viewed as dynamical systems on (T d ) n /S n .
Graphs of closed one-forms that lie in the level sets of the Hamiltonian are invariant under the flow (2.2). Since closed forms on (T d ) n /S n are given by ω x (p) = n i=1 c · p i + Du(x), p , for some c ∈ R d and u : (R d ) n → R periodic, we are led to the equation Qualitative properties of (2.1) and (2.2) are closely linked to the regularity properties of the solutions to (2.3).

The random variable approach
A standard method for studying mechanical systems with an infinite number of identical particles is to look at a probability measure encoding the positions of the particles. The evolution of a system is a curve in a space of probability measures. For problems with a finite number of particles, this measure is the empirical measure of the particles' positions. If the ambient space is compact, the space of probability measures on it is also compact. Compactness is particularly relevant for a variational theory such as the Weak KAM theory. On the other hand, the lack of a linear structure makes it more complex to introduce notions such as the derivative of a path.
An alternative approach consists of regarding the state of the system as a random variable. Each realization of this random variable represents the position of one particle. The evolution of the system is given by a trajectory in a space of random variables. The space of random variables is a vector space and has a natural Riemannian structure. Unfortunately, in contrast to the space of probability measures, non-trivial spaces of random variables are not locally compact. However, for symmetrical problems, random variables that have the same law represent an equivalent state of the system. Thus, the dynamics can be viewed as an evolution in the quotient space of random variables with respect to the equivalence relation of having the same law. This latter space is compact and isometric to the space of probability measures [28]. Hence, we can use compactness arguments.
In [26,27] the random variables approach is used by working in L 2 ([0, 1], R). Let P 2 (R) be the space of probability measures over R with finite second-order moment endowed with the 2-Wasserstein distance W 2 . Then, P 2 (R) is isometric to the set of monotone non-decreasing functions in L 2 ([0, 1], R). The lack of compactness of L 2 ([0, 1], R) is offset by the compactness of the set of monotone functions via Helly's selection theorem.
If d > 1, the random variables approach leads to a dynamical system on L 2 (I; R d ), where I = [0, 1] d . Unfortunately, unlike in dimension 1, there is no canonical isometry between P 2 (R d ) and some subset of L 2 (I; R d ). Hence, the methods used in [26,27] cannot be applied if d > 1. In particular, the existence of minimizing curves for both the discounted infinite horizon problem and the Lax-Oleinik semigroup were open until now. In this paper, we prove the existence of minimizers in the general case d ≥ 1. In [28], the authors use an alternative approach and work directly in P(T d ). Our techniques are more functional analytic in spirit, and use results from the calculus of variations in Hilbert spaces [33].

Mechanical systems with an infinite number of indistinguishable particles via random variables
Consider a mechanical system with an infinite number of identical particles. Assume that there is a one-to-one correspondence between particles and points in I = [0, 1] d . We encode the positions of the particles in a random variable M ∈ L 2 (I; R d ). Using the notation of [26,27], for each point z ∈ I, M z ∈ R d is the position of the particle "z" in the space.
Let L : L 2 (I; R d ) × L 2 (I; R d ) → R be the Lagrangian of the system. The associated dynamics is given by the Euler-Lagrange equation where the partial derivatives are in Fréchet sense, and (M, N ) ∈ L 2 (I; R d )×L 2 (I; R d ) is the initial configuration of positions and velocities. For H as in (1.4), the infinite-dimensional Hamiltonian system is then As in the finite-dimensional case, we need the notions of "periodicity" and "invariance under permutations" for the Lagrangian and Hamiltonian. Consider the subset of This set is a subgroup with respect to addition. A function F defined on Periodicity of the Lagrangian L in the spatial variable means that Let (X, F) and (Y, G) be measurable spaces, and M : ( Consider the set G of all bijective functions G : I → I such that G and G −1 are Borel measurable and that push-forward the Lebesgue measure λ 0 to itself. Then G, equipped with the composition operation, is a non-commutative group that plays the role of S n in the infinite-dimensional setting. Hence, invariance under permutations of the Lagrangian L in the infinite-dimensional setting is the invariance under the action of G: for all M, N ∈ L 2 (I; R d ) and G ∈ G. We call this property rearrangement invariance. If L is periodic and rearrangement invariant, the Euler-Lagrange equation (2.4) is a dynamical system on the d-infinite-dimensional symmetrical torus T d /G.
A thorough analysis of the symmetrical torus T d /G can be found in [26] (d = 1) and in [28] (d > 1). Here, we recall several important facts that we require for our analysis.
We endow T d /G with the induced metric dist weak defined as This distance satisfies all the axioms of a metric distance except the non-degeneracy, that We define an equivalence relation as follows: It is straightforward to see that ∼ is an equivalence relation. Define SS d as Remark 2.1. By the abuse of notation, we denote by M all equivalence classes of M .
Furthermore, for any continuous periodic function, F : L 2 (I; R d ) → R, the following assertions are equivalent: Finally, we set π : to be the natural projection that maps a function M to its equivalence class. Note that π is 1-Lipschitz.

Main assumptions
Here, we suppose that L :  N ), For c ∈ R d , let L c be as in (1.2). The Hamiltonian, H, associated with the Lagrangian, L, is given by (1.4). We refer to the second variable of the Hamiltonian, P , as the momentum variable. Differentiation with respect to the momentum variable is denoted by D p . Differentiation with respect to the first variable M is denoted by D x .
Assumptions i)-viii) yield that H is C 1 in Fréchet sense, strictly convex, and coercive. These duality statements can be found in [25], in the finite-dimensional case. Similar techniques apply to the infinite-dimensional case.

The discounted-cost infinite-horizon problem
In this section, we study the discounted-cost infinite-horizon problem. As is standard in Weak KAM theory, this problem can be used to build solutions to the cell problem [39].
Recall that a function taking values on R ∪ {±∞} is proper if it is not identically ±∞.
The set D − V (M ) (resp. D + V (M )) is the set of subdifferentials (resp. superdifferentials) at M . Let V ε be the discounted value function given by (1.3). Since the Lagrangian L c is rearrangement invariant and periodic in the spatial variable, so is the value function V ε .
We collect several elementary properties of the value function V ε in the following proposition. ii) For every ε > 0 the function V ε is Lipschitz continuous with Lipschitz constant independent of ε.
iii) For every ε > 0 the function V ε is semiconcave.
Proof. In the finite-dimensional case, these facts are standard and are discussed, for instance, in [25,3,4]. In the infinite-dimensional setting, the same methods can be applied without changes. Properties of the value function are examined, in the context of viscosity solutions, in [14,34,40,9,38,33].
Corollary 3.1. The superdifferential D + V ε is nonempty at every point M ∈ L 2 (I; R d ).
Besides, V ε is Fréchet differentiable on an everywhere dense G δ set.
Proof. A convex function on a Banach space has a non-empty subdifferential at every point where it is finite and continuous [37]. Moreover, if the Banach space is also a strong differentiability space [2], then every convex function defined on it is Fréchet differentiable on a G δ dense subset of its domain of continuity. L 2 (I; R d ) is a strong differentiability space (Theorem 1, [2]). Furthermore, V ε is semiconcave, finite and everywhere continuous. Accordingly, D + V ε (M ) = ∅ for all M ∈ L 2 (I; R d ) and V ε is Fréchet differentiable on a G δ dense subset of L 2 (I; R d ).
In [26], the authors proved that, in the one-dimensional case, when M is monotone non-decreasing, (1.3) admits a minimizer in H 2 loc ((0, ∞); L 2 (I)) that satisfies the Euler-Lagrange equation. Here, we establish the existence of minimizers on an everywhere dense G δ subspace of L 2 (I; R d ), for any d ≥ 1.
Next, we detail the proof of the main result of this section, Theorem 1.1.
Proof of Theorem 1.1. In [33], we studied the finite horizon optimal control problems in Hilbert spaces. We proved that at every point of differentiability of the value function, there exists a unique C 1 minimizer (Theorem 6.2, [33]). Since the infinite horizon problem can be seen as a finite horizon one, the existence of x * is a direct consequence of that result. It is also standard that minimizers solve the Euler-Lagrange equation (1.1) [38].

The infinite-dimensional weak KAM theory
In this section, we prove our main results: Theorem 1.2, Theorem 1.3, Corollary 1.1, and Corollary 1.2.
Closed one-forms on T d /G are given by DU + cχ I , for some periodic function U : L 2 (I; R d ) → R and some c ∈ R d [26,28]. Hence, the cell problem associated with (2.4) is where λ ∈ R. Moreover, as stated in Theorem 1.2, for every c ∈ R d there exists a unique number λ =H(c) such that (4.1) has a periodic rearrangement invariant viscosity solution U . In Proposition 4.2, we prove that this solution is a fixed point of the Lax-Oleinik semigroup, that is, 2) for any M ∈ L 2 (I; R d ) and t < t 1 . The case d = 1 was studied in [26,27], in a slightly weaker form in what concerns the Lax-Oleinik semigroup. Analogous results are available on the space of probability measures in [28].
Additionally, we show that U is semiconcave, and hence Fréchet differentiable on a G δ everywhere dense set (Proposition 4.3). Furthermore, at differentiability points M of U , the infimum in (4.2) is attained at a C 1 minimizer (Theorem 1.2). This issue was settled for d = 1 in [26] using different ideas, and the higher-dimensional case was not addressed there. A corresponding result on the space of probability measures can be found in [28].

The cell problem: existence of solutions and elementary properties
We begin by considering the limit as ε → 0 of the solutions V ε to (1.5). ii) The family of functions {U ε } has a uniformly convergent subsequence with a Lipschitz continuous limit U . Additionally, the family of functions {εV ε } has a uniformly convergent subsequence with constant limit depending on c: −H(c).
Remark 4.1. A priori, the constant limit of the convergent subsequence of the {εV ε } is not unique, and Proposition 4.2 is valid for any such limit and corresponding limit function U . However, it is simple to check that(1.7) implies the uniqueness of such a constant.
Hence,H(c) is uniquely determined by the vector c ∈ R d .
Proof. The family of functions V ε is equilipschitz (Proposition 3.1), thus, the family U ε is also equilipschitz.
The Lagrangian L c is rearrangement invariant hence V ε and U ε are also rearrangement invariant functions. By Proposition 2.1, we may identify U ε and V ε with functions on S d . Since S d is a compact metric space, U ε reach their minima that are 0. Furthermore, since they are uniformly Lipschitz, we obtain that {U ε } is bounded equicontinuous family of functions on the compact space S d . Therefore, by the Arzela-Ascoli Theorem, we conclude that it has a uniformly convergent subsequence. The limit U is also Lipschitz continuous.
From Proposition 3.1, we have that {εV ε } is a uniformly bounded and equicontinuous family of functions. Hence, by the Arzela-Ascoli theorem, we obtain that it has a uniformly convergent subsequence. The limit of this subsequence has Lipschitz constant 0, which is a constant function. Proof. Fix any M ∈ L 2 (I; R d ). We claim that for any x ∈ AC 2 ((0, t); L 2 (I; R d )) such that From (3.2), we have that, for every ε > 0, e −εs L c (x(s),ẋ(s))ds + e −εt V ε (x(t)).
Passing to the limit when ε → 0 and using Proposition 4.1, we obtain (4.3).
The previous inequality can be rewritten as . Thus, if we show that I, J → 0, we are done. Due to Proposition 4.1, I → 0 . Assumptions v)-vii) guarantee that L c is bounded by below. Since adding a constant to L c in J does not change the limit, we can assume that L c ≥ 0. Hence, because the sequence t 0 e −εns L c (x n ,ẋ n ) is bounded.
The function U enjoys properties analogous to the ones satisfied by V ε , namely:

Existence of an invariant subset
A trajectory x ∈ AC 2 loc (R; L 2 (I; R d )) is called a two-sided minimizer (or two-sided (U, c, L)calibrated curve) if for all −∞ < t 1 < t 2 < ∞.
We proceed by proving some preliminary lemmas. Define D t ⊂ T * L 2 (I; R d ) as the set of all points (M, P ) ∈ T * L 2 (I; R d ) for which there exists a solution (x(s), p(s)) of (4.5) with initial data (x(0), p(0)) ∈ D, and (x(t), p(t)) = (M, P ). In other words, D t is the image at time t of the set D under the Hamiltonian flow (4.5).
Since U is a value function, it is differentiable along the minimizing trajectory (Corollary 4.1, [33]). Therefore, D t ⊂ D, for any t > 0. Hence, since D is a graph, D t is also a graph, for all t > 0. Moreover, D t ⊂ D s for all s < t.
Let D t be the projection of the set D t onto the spatial component of the cotangent bundle T * L 2 (I; R d ), that is, all points M ∈ L 2 (I; R d ) such that (M, ∇U (M )) ∈ D t .
Since U is periodic and invariant under measure-preserving transformations, by Lemma 4.1, we have that U is also differentiable at M 2 . Furthermore, there exist G n ∈ G and Z n ∈ L 2 Z (I; R d ) such that M 1 • G n + Z n → M 2 in the strong L 2 sense. Denote by M n := M 1 •G n +Z n . Recall that the gradient of a convex function is continuous at all points where it is defined, see [2]. Then, we have that ∇U ( Because (M 1 , ∇U (M 1 )) ∈ D t , there exists a minimizing trajectory x such that (x(0), p(0)) ∈ D and (x(t), p(t)) = (M 1 , ∇U (M 1 )). Consider the trajectories y n (s) = x(s) • G n + Z n and q n (s) = p(s)•G n . Due to the rearrangement invariance and periodicity of the Hamiltonian H c , we have that (y n , q n ) solves (4.5) with terminal data (y n (t), q n (t)) = (M n , ∇U (M n )).
Let (y, q) ∈ C 1 [0, t]; L 2 (I; R d ) be the solution of (4.5) with terminal data (y(t), q(t)) = (M 2 , ∇U (M 2 )). Note that (y, q) is well defined since the existence of the solution for all times is guaranteed by the fact that the right-hand side of (4.5) is uniformly Lipschitz in (x, p).
The solutions (y n , q n ) to (4.5) have terminal data (y n (t), q n (t)). This data converges, in the strong sense, to the terminal data (y(t), q(t)) of another solution (y, q) of the same system. Therefore, by the stability of ODEs we obtain that y n (s) → y(s), q n (s) → q(s), uniformly in the interval [0, t]. But this means that y n (0) = x(0) • G n + Z n converges strongly to y(0), hence y(0) ∼ x(0). Then, by Lemma 4.1, y(0) is also a differentiability point of the function U . By the continuity of the gradient, we have q(0) = lim n→∞ q n (0) = lim n→∞ ∇U (y n (0)) = ∇U (y(0)).
This means that y is a minimizing trajectory starting at the differentiability point y(0). Therefore, (M 2 , ∇U (M 2 )) = (y(t), q(t)) ∈ D t or, equivalently, M 2 ∈ D t . Proof. Suppose M ∈ D t . Then, there exist points M n ∈ D t such that M n → M . Because M n ∈ D t , U is differentiable at M n and (M n , ∇U (M n )) ∈ D t . Furthermore, there exist minimizing trajectories x n such that (x n (t), p n (t)) = (M n , ∇U (M n )) and (x n (0), p n (0)) ∈ D. Since x n is a minimizer, the Lagrangian L satisfies the assumption viii), and U is a terminal cost function (as well as a value function), there exists a constant C depending on time t such that for all H ∈ L 2 (I; R d ) and all n. Because U is Lipschitz, the sequence {∇U (M n )} is bounded. Consequently, it has a weakly convergent subsequence with a limit P ∈ L 2 (I; R d ). Hence, by passing to the limit in (4.6), we obtain that for all H ∈ L 2 (I; R d ). Therefore, P belongs to the subdifferential D − U (M ). Due to the semiconcavity of U , the superdifferential D + U (M ) is non-empty. Consequently, U is differentiable at M and P = ∇U (M ) is its gradient.
By the continuity of the gradient, we have that ∇U (M n ) → ∇U (M ) in the strong L 2 sense. Next, solve (4.5) with terminal data x(t) = M, p(t) = ∇U (M ). We have that (x n (t), p n (t)) → (x(t), p(t)). Hence, x n (s) → x(s), p n (s) → p(s), uniformly in the interval [0, t] by the stability of solutions of ODEs. Furthermore, Thus, by passing to the limit, we obtain Consequently, x is a minimizing trajectory. Since U is differentiable at all points of the minimizing trajectory, except, possibly, at the starting point, we obtain that (M, ∇U (M )) ∈ D s , for all s < t. Therefore, M ∈ s<t D s , so D t ⊂ s<t D s . Now, we claim that s<t D s ⊂ D t . Suppose M ∈ s<t D s or, equivalently, (M, ∇U (M )) ∈ s<t D s . Then, for every 0 < s < t there exists a minimizing trajectory x s (τ ) such that (x s (s), p s (s)) = (M, ∇U (M )). Let (x, p) be the solution of (4.5) with data (x(t), p(t)) = (M, ∇U (M )). By the uniqueness of the solution to (4.5), we have that (x s (τ ), p s (τ )) = (x(τ + t − s), p(τ + t − s)), for any s < t. Therefore, the trajectory x is a minimizer for any starting point (x(t − s), p(t − s)). Consequently, (x(2t − s), p(2t − s)) ∈ D t . So   Proof. Since all the sets D t are graphs, the statement in the lemma is equivalent to From Lemma 4.3, we obtain that D ∞ = t>0 D t . The sets {D t } t>0 are closed nested sets.
Consider projections of A t onto SS d through the projection operator π (see (2.6)).
Due to Corollary 4.1, the sets D t contain only full equivalence classes with respect to the equivalence relation ∼. Since the sets D t are closed, the sets A t are also closed. Additionally, they are compact, because SS d is compact. Accordingly, they have a nonempty compact intersection Therefore, Now, we have all the prerequisites to prove our next main result, Theorem 1.3. The projection operator π is continuous, and the set A ∞ is compact and hence closed. Therefore, the set D ∞ is closed. Since the gradient of U is continuous on the set of differentiability, we obtain that The existence of weakly invariant minimizing measures on the tangent space of P(T d ) was shown in [28]. Subsequently, in [9], the author established the existence of invariant (or strongly invariant) minimizing measures on T L 2 ([0, 1]). Here, we settle the remaining question, namely the existence of strongly invariant measures in T L 2 (I; R d ) for all d ≥ 1. For that, we extend the methods introduced by Fathi in [16] to the infinite-dimensional setting.
Let A be the Borel σ-algebra of the subsets of A ∞ . Define Note that D is a σ-algebra of subsets of D ∞ . For every C ∈ D, set B is a σ-algebra of subsets of Ω. ii. the space of probability measures on (Ω, B) is a narrowly compact space. Proof.
i. Let f : A ∞ → R be a continuous function. For (M, N ) ∈ Ω, let F f (M, N ) = F f (M, −D p H c (M, ∇U (M ))) = f (π(M )). Then, F f is rearrangement invariant and continuous. Define q is a linear bounded functional acting on continuous functions f : A ∞ → R. Since A ∞ is compact, the Riesz Representation Theorem yields the existence of a Borel measure ν on A ∞ such that Consider the measure µ(B C ) = ν(π(C)), C ∈ D ∞ . ii. Suppose {µ n } are probability measures on (Ω, B). Consider the sequence of measures ν n given by (4.8). Since, ν n are supported on a compact set A ∞ , they form a narrowly precompact sequence. Hence, there exists a measure ν ∞ such that ν n → ν ∞ narrowly. Now, define µ ∞ via (4.8). Then, it is straightforward to verify that µ n converges to µ ∞ narrowly (tested against continuous rearrangement-invariant functions).  The previous identity gives that µ ∞ is a minimizing measure.