Random Lift of Set Valued Maps and Applications to Multiagent Dynamics

We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.


Introduction
The mathematical analysis of complex systems with multiple interacting agents has recently obtained a prominent importance in the interdisciplinary research involving applied mathe-matics.The main fields of application encompass social dynamics, e.g., pedestrian dynamics, social network models, opinion formation, infrastructure planning, ranging from financial markets to engineering (unmanned vehicles, energy distribution on networks), from big data analysis (including tools of artificial intelligence) to life sciences (e.g.flocking, spreading disease models).
A characterizing feature of the analysis of multiagent systems, like e.g., cell populations, fish swarms, insect colonies, human crowds, bird flocks, is given by the fact that the collective behaviour is deeply influenced by complex interactions that usually arise among the subjects (agents).The interaction between the agents may range from the simplest, e.g., attraction/repulsion effects or collision avoiding to more complex ones, involving also penalization of overcrowding/dispersion, or further state constraints on density of the agents.Summarizing, the description of the collective behaviour cannot be reduced to the simple superposition of individual behaviours neglecting the influences of the mutual interactions.
The second relevant features of multiagent system is given by the fact that the number of involved agents is huge, making impossible in practice to track the evolution of every single agent, therefore only some aggregated information on the system will be available.We refer to this as macroscopic description of the system.A great simplification, quite commmon in all the multiagent systems models in the main domains of applications, is to consider the influence that each agent receive due to the presence of all the others is a function of the same aggregated information macroscopically describing the system.This feature is often called indistinguishability of the agent: in the sense that an external observer can detect only the aggregated information generated by all the agents, and also each agent perceive the influence of all the others not as single individuals but only through the variation of the aggregated information, thus reducing the complexity of the system to the analysis of the evolution of these aggregated information.
For instance, if we assume that each agent is influenced just by the position of the others, for a single agent what matters will be just the total number (or the fraction of the total number) of agents present at the given times in each point of the space, regardless of the label carried by each agent.In this case, the aggregated information describing the system will be just the distribution in space of the agents.
Even if the range of models and situations where multiagent models can be applied is very wide, a large part of them shares the two above features outlined.Therefore a natural question is whether is possible to provide a common framework to deal with these problems.The framework should be sufficiently flexible to cover a wide range of microscopic agent evolutions and in the choice of the macroscopic quantities describing the state of the system.For instance, beside the Euclidean case, the underlying spaces where the microscopic agents move can be for instance in metric graphs -as in network traffic models.Furthermore, the admissible evoutions of each agent can be of different nature with respect to solutions to differential inclusions/controlled equations: indeed, the agent may simply freely move in order to minimize some personal cost, as common in mean field game theory.
Moving from these considerations, in this paper we build a general theory introducing and studying an abstract two-scale evolution: we adopt a Lagrangian point of view by assigning to each point of the underlying state space a family of microscopical admissible trajectories depending by a given macroscopical evolutions (considered as an input).The produced assignment, modeled by a suitable set-valued map, is then weighted according to the initial distribution of the agents on the starting points, and to the weigths assigned to each admissible evaluation.This produces again a macroscopical evaluation as an output, which depends on the macroscopical evolutions chosen at the beginning.
The presence of a fixed point in this construction, i.e., the correspondence between input and output, will provide the existence of an admissible macroscopical evolution where the microscopical evolutions of the agents and theit resulting macroscopical superposition are coherent.
The construction outlined so far leads to the following natural question, which basically constitute the aim of the investigation in this paper.
(1) Which properties on the initial microscopical assigment ensure the existence of a macroscopical evolution starting from an initial distributions of agents?
(2) Which properties on the initial microscopical assigment (such as closedness of the graph, compactness of the images...) are transferred to the set valued map of the fixed point, i.e., to the set valued map of the admissible macroscopic evolutions?
(3) Derive special results for the multiagent systems evolving in an underlying finitedimensional space.
The proposed theory improves the previous ones in the following aspects: • by extending the existence results for macroscopical evolutions, previously proved in [9] and [10] for an underlying Euclidean space, to general separable metric spaces (see Proposition 2.9 and Proposition 2.12); • in the case of microscopical evolution driven by a differential inclusion, by removing the assumption of convexity, used in [22] and [15], to prove a Filippov-type estimate on the macroscopical evolutions (see Theorem 3.10), more in the spirit of the finite-dimensional version of such result.By standard techniques, this leads to a linearization and a relaxation results for macroscopical evolutions (see Proposition 3.11 and Proposition 3.13).
We then apply our new approach to some relevant example, like • constrained mean field games equilibria, where the microscopical evolutions are chosen to be the minimizers of a suitable integral functional subject to state constraints (see Proposition 4.2): this concept was originally introduced in [6], and our approach derives immediately the main result of [6] as a consequence of the proposed general theory; • Cauchy-Lipschitz theory for continuity equations, where we show that some of the results of the theory proposed in [5] for a particular concept of solution of a continuity equation driven by a differential inclusion, different from the one constructed in Sect.3, can be quickly derived also from our setting (see Sect. 5).
The paper is structured as follows.In Sect. 2 we will precise the abstract construction outlined in this Introduction.We will then apply the construction to three main frameworks: in Sect. 3 to multiagent systems moving in the Euclidean space and whose microscopic evolution is provided by a differential inclusion, in Sect. 4 to mean field games in presence of state constraints, and in Sect. 5 to the Cauchy-Lipschitz framework for controlled continuity equations.We refer the reader to the Appendix for notation and some technical results.

Random Lift of Set Valued Maps
We precise in this section the construction outlined in the Introduction.Notice that the notion of random lift of set valued maps presented here generalizes the one introduced in [9] and [10], extending the theory stated for the Euclidean space to metric spaces, and so encompassing more possible applications.

General Construction
Consider three separable metric spaces X, Y , Z, which we are going to term in the following way in order to keep the link with the multiagent system: • X is the space of initial states; • Y is the space of microscopic evolutions of the agent; • Z is the space of observable macroscopic evolution.
We consider then a set valued map S : Z × X ⇒ Y , which associates a microscopic evolution to each macroscopic evolution and state.We set also S z (x) = S(z, x), thus defining S z : X ⇒ Y for every given z ∈ Z.
Roughly speaking, in a multiagent system interpretation, the set valued map S associates to each pair (x, z) ∈ X × Z the set of admissible microscopic evolutions starting from the initial state x ∈ X provided that the observable macroscopic evolution is described by z ∈ Z. Choosing the state space and the microscopic evolution space in form of a product space take in account different population of agents, with possibly different evolution types.
In many cases, the set valued map S : Z × X ⇒ Y turns out to be not adequate to describe the system, as customary in mean-field constructions.Among the possible motivations, we single the following ones: • modeling situations where only a probabilistic knowledge of the initial state is available (due to, e.g., noise in the measurements, systems with incomplete information...); • modeling multiagent systems, where only a statistical aggregated description of the evolution is available (related to the information that each agent can extract from the system about the others).
Inspired by this problem, starting from the set valued map S : Z × X ⇒ Y , we define its random lift w.r.t.X to be the set valued map : Z × P(X) ⇒ P(X × Y ), with (z, μ) := {η ∈ P(X × Y ) : η(graph S z ) = 1, and pr 1 η = μ}, where pr 1 (x, y) = x for all (x, y) ∈ X × Y and r θ denotes the pushforward measure of the measure θ w.r.t. a Borel map r, which will be defined in Sect.A.1.We will also write z (μ) = (z, μ).
Given η ∈ z (μ), for η-a.e.(x, y) ∈ X × Y we have that y ∈ S z (x), thus the measure η charges mass only on the subsets of admissible microscopic evolutions under macroscopic observable evolution described by z.
According to the Disintegration Theorem (see e.g.Theorem 5.3.1 in [1]), we write η as the product measure η = (pr 1 η) ⊗ η x = μ ⊗ η x , where {η x } x∈X is a Borel family of probability measures, uniquely defined for μ-a.e.x ∈ X, and pr 1 η is the pushforward measure of η w.r.t. the Borel map pr 1 .Informally, • the measure μ represents the weights assigned to the initial conditions in X, • η x is a probability measure concentrated on the set S z (x) of the admissible evolutions starting from x which describes how the mass initially present at x is splitted on the evolutions from x under the macroscopic observable evolution z.
After having constructed the random lift : Z × P(X) ⇒ P(X × Y ), which describes the admissible evolutions in a probabilistic form, we introduce an evaluation map E : P(X × Y ) → Z, which plays the same role of an observability relation, expressing which information from the probabilistic evolutions can be actually catched from the system (and affects the set of microscopic evolutions).We then define the set valued map ϒ : and write ϒ z (μ) = ϒ(z, μ).The set ϒ z (μ) describes the observable macroscopic evolutions starting from μ under the observable macroscopic evolution described by z.This naturally leads to a fixed point construction: indeed, we define the set of admissible observable macroscopical trajectories starting from μ as the fixed point of the set valued map z → ϒ z (μ), i.e., Indeed, there is coherence between the (weighted) microscopic evolution and the macroscopic observed evolution if and only if z ∈ ϒ(z, μ).
The set valued map A : P(X) ⇒ Z expresses all the admissible macroscopic observable evolution with respect to the weight assigned on the initial state.
In this way, the basic issues raised in the Introduction turn out to establish which properties on the set valued map S : Z × X ⇒ Y ensures that A (μ) = ∅ and are possibly inhereted by A : P(X) ⇒ Z.
Let X, Y , Z be complete separable metric spaces, and S : Z × X ⇒ Y be a Borel set valued map.Given z ∈ Z, we denote by S z : X ⇒ Y the set valued map x → S(z, x).In particular, (z, x, y) ∈ graph S if and only if (x, y) ∈ graph S z .Definition 2.1 (Random lift of a general set valued maps) The set valued map : Z × P(X) ⇒ P(X × Y ) defined as (z, μ) := {η ∈ P(X × Y ) : η(graph S z ) = 1 and pr 1 η = μ}, where pr 1 (x, y) = x for all (x, y) ∈ X × Y , will be called the random lift of S with respect to X.Given z ∈ Z, we denote by z : P(X) ⇒ P(X × Y ) the set valued map μ → (z, μ), which will be called the random lift of S z (•).Definition 2.2 (Evaluation for random lift) Given p ≥ 1, a map E p : P p (X × Y ) → Z will be called an evaluation for the random lift (•).We say that an evaluation and η In the following definition we characterize the set of fixed-points associated to the random lift and its evaluation map.Definition 2.3 Given an evaluation E p for the random lift (•), we define the set valued map ϒ Ep : Z × P p (X) ⇒ Z by setting for all z ∈ Z and μ ∈ P p (X) and the set valued map A : P p (X) ⇒ Z by Given z ∈ Z, we denote by ϒ z Ep : P p (X) ⇒ Z the set valued map μ → ϒ Ep (z, μ).When p is clear from the context, we will simply write ϒ z (•).

Properties of the Random Lift
Our aim is to derive relevant properties of the random lift of a set valued map S(•) starting from corresponding properties of S. Proposition 2.4 Let X, Y , Z be complete separable metric spaces, and S : Z × X ⇒ Y be a Borel set valued map.Denote by (•) its random lift.
(1) the images of (•) are always either empty or convex images w.r.t. the linear structure of (C 0 b (X × Y )) , even if S(•) has nonconvex images.If Z is a convex subset of a linear space and E p is an affine evaluation, we have that ϒ Ep (•) has convex or empty images.
(2) for every (z, μ) ∈ Z × P(X) and any Borel set A ⊆ X, it holds

) S(•) has closed graph if and only if (•) has closed graph;
(5) given z ∈ Z, suppose that for every compact K ⊆ X the set graph  with uniformly integrable p-moments, where the metric on X × Y is defined by In particular, the restriction z |Pp (X) of z (•) to P p (X) takes values in P p (X × Y ).
(2) if (i.)-(ii.)holds, then for every relatively compact K ⊆ P p (X), the set z (K ) is relatively compact in P p (X × Y ).
( In particular, if S z (•) is upper semicontinuous with compact images and at most linear growth, then (1)-( 2)-(3) holds.Proof 1.Take μ ∈ K , η ∈ z (μ) and let C > 0 and ( x, ȳ) ∈ graph S z as in the linear growth condition for S z .For r > 0, define s Due to the uniform integrability of the p-moments of K , given ε > 0 there exists s ε > 0 such that for all s ≥ s ε and μ ∈ K it holds thus, recalling the definition of s(r), we conclude that there exists r ε = p √ C p + 1(s ε + 1) > 0 such that for all r > r ε and η ∈ z (K ) it holds i.e., η ∈ z (K ) has uniformly integrable p-moments.2. By Proposition 7.1.5 in [1], K has uniformly integrable p-moments and it is tight.According to item (1), we have that z (K ) has uniformly integrable p-moment.Since Proposition 2.4(5) implies that z (K ) is relatively compact in P(X), and so it is tight for Proposition 5.1.3 of [1], we conclude that z (K ) is relatively compact in P p (X × Y ) again by Proposition 7.1.5 in [1].
3. By Proposition 2.4(4), we have that graph z is closed in P(X × Y ).Since W pconvergence implies narrow convergence, graph z ∩ P p (X) × P p (X × Y ) is W pclosed.By item (1), we have that z (μ) ⊆ P p (X × Y ) for all μ ∈ P p (X), and so is W p -closed.By item (2), z (μ) is relative compact for all μ ∈ P p (X), and therefore the W p -closure of the graph implies that z (μ) is compact in P p (X × Y ).By continuity, E p ( z (μ)) is compact in Z.If E p sends closed sets to closed sets, then (Id Pp (X) , E p )(•, •) sends closed sets to closed sets, hence is closed.
The last assertion follows from the well-known facts that compact-valued upper semicontinuous maps between metric spaces send compact sets to compact sets (see e.g.Theorem 2.1 in [12]) and that an upper semicontinuous multifunction has always closed graph (Proposition 1.4.8 of [3].
The following results can be interpreted as a lifted version of Filippov's Implicit function theorem.
If X, Y are σ -compact, then the compactness of the images of S z (•) can be replaced by just closedness.
Proof By considering the sequence constantly equal to the function f (x, y, x , y ) = d W (g(x, y, x ), h(x, y, x , y )) ≥ 0, we have that g(x, y, x ) ∈ {h(x, y, x , y ) : y ∈ S z (x)} amounts to have the existence of ȳ ∈ S z (x) such that g(x, y, x ) = h(x, y, x , ȳ ) i.e., f (x, y, x , ȳ ) ≤ 0. Thus the assumptions of Lemma 2.6 are satisfied.Since f ≥ 0, the conclusions provide f (x, y, x , y ) = 0, thus

Theorem 2.8 (Uniform continuity) Suppose that S(•) has compact images and that there exists a concave continuous function ω
where d H denotes the Hausdorff distance.Then (1) z (P p (X)) ⊆ P p (X × Y ) for all z ∈ Z; (2) for any K relatively compact in P p (X) and z ∈ Z, we have that z (K ) is relatively compact in P p (X × Y ).In particular, the restriction of z (•) to P p (X) has compact images in

uniformly continuous with respect to the
Hausdorff distance: where In is concave and upper semicontinuous with nonempty domain, By taking s 1 = s 2 = 0 in the above definition, we deduce q ≥ 0.Moreover, by taking s 1 = 0 and s 2 → +∞ we have m 2 ≥ 0, and similarly we have m 1 ≥ 0. According to Lemma 2.5, to prove (1) and ( 2) it is sufficient to show that S z (•) has at most linear growth and that for every compact K ⊆ X the set graph Fix x ∈ X and ȳ ∈ S z ( x).Let y ∈ S z (x), and choose y ∈ S z ( x) such that d S z ( x) (y) = d Y (y, y ).The existence of such y follows from the compactness of S z ( x), which also ensures the boundedness of diam(S z ( x)) := sup and the linear growth follows by taking C = (m 1 + q + diam(S z ( x))).
Since S z (•) has compact images and it is continuous, we can apply Theorem 2.1 in [12] which ensures that for every compact K ⊆ X the set graph S |K is compact in X × Y .Hence (1) and ( 2) are proved.
we have where we used the fact that pr 13 ξ = π .For any (m 1 , m 2 , q) ∈ R 3 such that m 1 r 1 + m 2 r 2 + q ≥ ω S (r 1 , r 2 ) for all r 1 , r 2 ≥ 0, we have recalling that m 1 , m 2 , q ≥ 0. By taking the infimum on (m 1 , m 2 , q) and recalling the concavity and the continuity of ω S we obtain Since π is optimal between μ and μ , we have . Thus, by reversing the roles of μ , μ, we have the desired uniform continuity of z (•) with modulus ω (•).In the Lipschitz case, we apply the above arguments by replacing ω S (•) with ω S z (r) := r • Lip S z and taking z = z .
Concerning (4), we have that if ω Ep is a modulus of continuity for E p , by the properties of Hausdorff distance the modulus of continuity for ϒ(•) is ω Ep • ω .

Fixed-Point Theorems for Random Lift
Proposition 2.9 (Fixed point -Case I) Let p ≥ 1, and Z be a subset of a linear space.Suppose that • (z, x) → S z (x) is upper semicontinuous with nonempty compact images and at most linear growth;

μ). Moreover, if E p maps closed sets to closed sets we have that the set valued map
defined on C with values on the subsets of Z, is upper semicontinuous.
Proof We prove (2) since (1) follows from (2) by taking G(z, μ) ≡ K.Under the assumptions on K, according to [20] and [25], there exists a locally convex topological vector space L , a subset L K ⊆ L and an homeomorphism h : where L K will be endowed with the topology inherited by L .Set empty compact images and closed graph since h(•) is an homeomorphism, G has closed graph by assumption and ϒ Ep has closed graph and compact images by Lemma 2.5.Moreover, it has convex images since h(•) sends convex sets to convex sets, and the images are all contained in a common compact set.By Kakutani-Fan-Glicksberg fixed point theorem (see e.g.Theorem 13.1 in [24]), this set valued map admits a fixed point, i.e., there exists , which concludes the first part of the proof.
According to Proposition 1.4.8 in [3], in order to prove the upper semicontinuity of (2.2), it is enough to show that the graph is closed.Recalling the definition of A (μ), we have to study the closedness of the following graph: whose domain is the closed set C , and whose images are contained in the compact set K. This is obvious using once again Proposition 1.4.8 in [3], since we already know that the map (z, μ) → ϒ z Ep (μ) ∩ G(z, μ) is upper semicontinuous with compact values and closed domain.
When there are available more refined estimates on the modulus of continuity of S(•), it is possible to improve the above result.To this aim, we will use the following parametric variant of Nadler's Fixed Point Theorem for contractive set valued maps.Theorem 2.10 (Parametric Fixed Points theorem for contractive set valued maps) Let X, Y be complete metric spaces, 0 < L < 1, and G : X × Y ⇒ X be a set valued map with compact nonempty images such that Then for every (z, w) ∈ X × Y, there exists z ∈ X such that z ∈ G(z, w) and If moreover there exists a modulus ω(•) such that we have that the set valued map Fix G : Y ⇒ X defined by has closed graph with nonempty images, and it is uniformly continuous with Proof The proof of the first part is classical, and we refer to Lemma 4.2 of [16] and Lemma 1 of [21].
In order to prove the closedness of the graph of Fix G (•), we consider {(z n , w n )} n∈N ⊆ X × Y with z n ∈ Fix G (w n ) for all n ∈ N and z n → z, w n → w.In particular, we have z n ∈ G(z n , w n ), and therefore For the last assertion, we take z ∈ Fix G (w 1 ), i.e. z ∈ G(z, w 1 ).Thanks to the first part, we know that there exists z ∈ Fix G(w 2 ) such that Reversing the role of w 1 and w 2 , we get, for z ∈ Fix G (w 2 ), Concerning the compactness of the images of the fixed point map, we will use the following result, providing the existence of fixed points for the pointwise limit G of sequence of set valued map {G i } i∈N , where G i admits fixed points for all i ∈ N. Theorem 2.11 Let (X, d X ) be a complete metric space, {G i : X ⇒ X} i∈N be a sequence of Lipschitz set valued maps with compact nonempty images, pointwise converging to G ∞ : X ⇒ X. Suppose that there exists 0 < α < 1 such that Lip G i < α for all i ∈ N. Then any Proof See Theorem 9 in [23].
We are ready to show an improved version of the fixed point argument in our situation when good estimates on S(•) are available.Proposition 2.12 (Fixed point -Case II) Suppose that E p is a 1-Lipschitz evaluation, S(•) has compact images and there exists a continuous concave function ω

compact and nonempty images, and it is uniformly continuous with modulus ω
Proof By setting X = Z, Y = P p (X), G = ϒ Ep , according to item (4) of Theorem 2.8 we have that where ω (r 1 , r 2 ) = r 1 +ω S (r 1 , r 2 ).In particular, taking μ = μ and then z = z , and recalling the assumption on ω S , we have Thus we can apply Theorem 2.10, obtaining that A (μ) = Fix G (μ) has closed graph, nonempty images, and it is uniformly continuous with modulus By applying Theorem 2.11 to the constant sequence of set valued maps G i (z) = ϒ z (μ), we obtain that if {z n } n∈N ⊆ A (μ) is a sequence of fixed points, i.e., z n ∈ ϒ zn (μ) for all n ∈ N, then it admits a subsequence {z

Application to Multiagent System Modeling
We are now ready to apply the constructions of the previous sections to the multiagent system model, as outlined at the beginning of the Introduction.
Most of the models of multiagent systems shares the following main assumptions: • indistinguishability: at each instant of time, each agent cannot distinguish between agents sharing the same position in space; • nonanticipativity: at each instant of time t ∈ I , the evolution of each agent is affected only by information retrieved at s ∈ [a, t].
Those assumptions describes the information that each agent can extract from the environment about all the others.
. Given a Lipschitz continuous set valued map F : I × P p (X) × X ⇒ X with convex, compact, and nonempty images, and μ ∈ P p (X), we have that μ = {μ t } t∈I ⊆ P p (X) is an admissible trajectory starting from μ for the system governed by the microscopical dynamic F if there exists a measure η ∈ P(X × I (X)) concentrated on pairs Given η ∈ P(X × I (X)), we set e I η = {e t η} t∈I ⊆ P(X).
Given t ∈ I , we consider the Lipschitz continuous map e t : X × I (X) → X defined as (3.2) (trivially, e t is linear and Let η ∈ P(X × I (X)) be representing the effective evolution of the agent.Given any Borel set B ⊆ R d , we have that e −1 t (B) ⊆ X × I (X) represents η-a.e. the pairs initial condition -trajectory of the agents that at time t are inside B. Since they are not distinguishable, we assign to B the total mass that they are carrying, i.e., η(e −1 t (B)).Set μ t = e t η ∈ P(X), this quantity amounts to be precisely μ t (B).Thus μ t represents the aggregate information obtained identifying the agents occupying the same position at time t ∈ I .Therefore we obtain a curve μ = {μ t } t∈I ⊆ P(X).By Disintegration Theorem, we can disintegrate η = μ t ⊗ η t,x , where {η t,x } x∈X is a family of Borel probability measures on X × I (X), uniquely defined for μ t -a.e.x ∈ X, representing the (weighted) trajectories of all the agent concurring at the position x at time t .
According to the indistinguishability property, the trajectory of each agent can be affected only by μ = {μ t } t∈I , i.e., F depends only on e I η := {e t η} t∈I (and not on the full η).By an abuse of notation, we will use e I η also to denote the function t → e t η from I to P(X).
The nonanticipative property establishes that if e s η (1) = e s η (2) for all s ∈ [a, t], then the influence on the trajectories of the agents is the same up to time t .Moreover, in the simplest case, we can have just a dependence on the instantaneous current state (see property (F5) and (F5 + ) later).
In this section we will provide some properties of the set of admissible trajectories in the space of measures.The existence of admissible trajectories will be proved by a fixed point argument in terms of random lift.The previous considerations motivate the following definition.
X be a finite-dimensional real space, Y = I (X), Z = C 0 (I ; P p (X)), endowed with their usual distances.Given a set valued map F : I × Z × X ⇒ X, which will be called the microscopic dynamics of the system, we define and, given θ ∈ Z we set ϒ F,θ I : P p (X) ⇒ Z to be ϒ F,θ I (μ) = ϒ F I (θ , μ).Finally, we define A F I : P p (X) ⇒ Z to be the set valued map given by Assumptions on F .In the situation of Definition 3.3, we can consider the following assumptions (F1 − ) F (•) has nonempty compact images.(F1) F (•) has nonempty compact convex images.(F2) There exists r(•) ∈ L 1 (I ) such that for all θ = {θ t } t∈I ∈ Z and for a.e.t ∈ I it holds (F3) For every θ ∈ Z, we have that t → F (t, θ , x) is measurable for all x ∈ X, and x → F (t, θ , x) is continuous for all t ∈ I .
Proof See Appendix.
Since E p is 1-Lipschitz (hence uniformly continuous), and ω S is concave, we can combine Lemma 3.5 and Theorem 2.8 to obtain the following result.• the set valued map ϒ F I : Z × P p (X) ⇒ Z has compact images and it is uniformly continuous, more precisely has compact images and it is uniformly continuous with the same modulus, i.e., Proof We select a < τ ≤ b such that From Corollary 3.6, we have that (θ , μ) → ϒ F [a,τ ] (•) has compact images and it is uniformly continuous with modulus ω ϒ (r All the results follow from Proposition 2.12. Lemma 3.8 (Convex structure of C 0 (I ; P p (X))) In the situation of Definition 3.3, given any closed convex subset K ⊆ Z endowed with the topology induced by Z, we have that is continuous.
Proof We prove just the first point, since the second one is obtained applying the function E p to K .Let C > 0 be such that 4 (C − 1) p > 1 + 6 m p ( μ) , and τ ∈ I be such that this choice is always possibile since, according to the integrability of r(•) on I , the left hand side vanishes as τ → a + .Define u : [0, τ ] × [a,τ ] → R by u(t, γ ) := 1 + 2|γ (t)| p .We will write u γ (t) = u(t, γ ), and set S = E p (K ), where K is defined as ) : e a η = μ, and for η-a.e.
Step 1: Trivially, K ε Y is closed and, since K ε X is compact, by Ascoli-Arzelà Theorem we easily obtain the compactness of K ε Y in C 0 (I ; X), and so the compactness of Hence K is tight and, thanks to Theorem 5.1.3 of [1], is relatively compact in P(X × [a,τ ] ).
We prove that the p-moment is uniformly integrable on K .We start noticing that for every η ∈ K , it holds Let ε > 0. Since μ ∈ P p (X), it is possible to find r ε > 0 such that if r > r ε it holds (X\B Set r 1 := p 4r p + 3 2 from (3.3) we know that, for η-a.e.
and so, coming back to (3.4), (X× Thus K is relatively compact in P p (X × [a,τ ] ).Finally, we prove that it is closed.Let {η (n) } n∈N ⊆ K and η ∈ P p (X × [a,τ ] ) such that η (n) → η in P p (X × [a,τ ] ).In particular, we have μ = e a η (n) → e a η, so e a η = μ.Furthermore, for every (x, γ ) ∈ supp η there exists a sequence {(x n , γ n )} n∈N converging to (x, γ ) and such that (x n , γ n ) ∈ supp η (n) for all n ∈ N. Thus by passing to the limit in γ n (a) = x n , we obtain γ (a) = x.Moreover, since γ → u γ (t) is continuous for all t ∈ [a, τ ], we have u γ (t) ≤ 3 2 u γ (a) by passing to the limit in u γn (t) ≤ 3 2 u γn (a) as n → +∞.Finally, since for all a ≤ s 1 ≤ s 2 ≤ τ Step 2: The set K is convex and non-empty in P p (X × Y ).Consequently, the set S is nonempty, convex and compact in C 0 ([a, τ ]; P p (X)).

Proof (of Step 2):
The compactness of S follows from the continuity of E p and from the compactness of K , whereas the convexity of S will follow from the convexity of K and recalling that E p is affine.We just have to prove that K is convex.To this aim, given η (i) ∈ K , i = 1, 2, and λ ∈ [0, 1] we have that supp(λη (1) + (1 − λ)η (2) ) ⊆ suppη (1) ∪ suppη (2) and e t (λη (1)  (2) , which immediately imply the convexity from the definition of K .Finally, it is easy to see that K = ∅ and S = ∅, since the constant curve θ = {θ t } t∈[a,τ ] with θ t ≡ μ, which can be written as the E p -image of a measure supported on constant curves, belongs to S .

Filippov's Theorem and Its Consequences
We provide now Filippov-like estimates and their consequences.The first one improves previous results obtained in [22] and [15], by removing the assumption of convexity.

w(γ )(s)) by the compactness of the images of G μ (n)
. By Filippov's Theorem (see e.g.Theorem 7.1 in [11]), there exists a Leb ⊗ Bor(X × Y )-measurable map v (n)  s, x, γ ) is a minimizer of the distance of w(γ )(s) from the elements of μ (n) (s, x, γ ).We will write v (n)  γ (s) in place of v (n) (s, x, γ ) noticing that there is no dependence on x.Define P n : X × Y → X × Y by setting P n (x, γ ) = ( γ (0), γ ) where In particular, for any γ ∈ AC(I ; X) we have Set In particular, we have η n+1 = P n η 0 and Similarly .
For any (x 0 , γ 0 ) ∈ X × Y such that γ 0 (0) = x 0 and γ 0 ∈ AC(I ; X), consider the sequence Thus, by taking the L p η 0 -norm in (3.5), we have According to Lemma 6.3, we have the convergence in L ∞ (I ) of the series ∞ i=1 a i which provides in particular • the uniform convergence of {μ (n) } n∈N to a limit denoted by μ ∞ in C 0 (I ; P p (X)); • the strong convergence of {P n } n∈N to a limit denoted by Applying again Lemma 6.3 to (3.5) with t = T , we obtain furthermore s ,γn(s)) ( γn (s)) ds = 0, by choosing a suitable subsequence, such that γn k (s) → γ∞ (s) for a.e.s ∈ I , we obtain by Dominated Convergence Theorem Integrating the previous relation w.r.t.η 0 , and recalling that η ∞ = P ∞ η 0 , we have We conclude that for η ∞ -a.e.
Hence, according to Lemma 6.3, taking b n (•) ≡ 0, and noticing that a 0 (0) = 0 we have The last assertion follows from theorem.
The first consequence of Theorem 3.10 is the following linearization result.The name is justified by the fact that this result allows to construct an admissible macroscopic trajectory having a contact of order greater than one with a weighted superposition of straight lines.Proposition 3.11 (Linearization) Let I be a compact interval of R, k(•) ∈ L 1 (I ), F : I × P p (X) × X ⇒ X be a set valued map with compact and nonempty images.Assume that F (•, μ, x) is measurable for every x ∈ X, μ ∈ P p (X) and that for a.e.t ∈ I we have Let μ ∈ P p (X), and v(•) ∈ L p μ (X) be a Borel selection of F (a, μ, •).Then there exists a measure η ∈ P p (X × Y ) such that

e. t ∈ I ;
• set μ = e I η, we have that μ is Lipschitz continuous, and there exists μ ∈ A F I (μ) satisfying In particular, for any locally Lipschitz map V : P p (X) → R, we have that if {t i } i∈N ⊆ I \ {a} is a sequence converging to a, then in the sense that if one of the two limits exists, then also the other exists and they agree.
We prove the Lipschitz continuity of μ by noticing that By Theorem 3.10, there exists an admissible trajectory μ ∈ A F I (μ) and η ∈ μ,F I (μ a ) such μt = e t η for all t ∈ I and , where we used Hölder inequality and the fact that for μ-a.e.x, we have from the Lipschitz hypothesis on We divide by s − a and integrate w.r.t.s on [a, t].Thus The right-hand side term belongs to L p μ due to the assumptions on v(•).Thus by Dominated convergence theorem and the last term vanishes when t → a + .
To prove the last assertion, we take a locally Lipschitz function V , and call C a the Lipschitz constant in a neighborhood of a.We notice that, recalling that μ a = μa = μ, we have for i sufficiently large which tends to 0 as t i → a + , as proved before.
Corollary 3.12 (Initial velocity set) × X ⇒ X be a set valued map with closed and nonempty images.Assume that F (•, μ, x) is measurable for every x ∈ X, μ ∈ P p (X) and that for a.e.t ∈ I we have Let μ ∈ P p (X), and v(•) ∈ L p μ (X) be a Borel selection of F (a, μ, •).Then there exists an admissible trajectory μ = μt } t∈I ∈ A F I (μ) such that ∂ t μt t=a = div(vμ), in the sense of distributions, i.e., for all ϕ ∈ C 1 c (X) Proof We construct η, μ and μ as in the proof of the previous Proposition.Given any ϕ ∈ C 1 c (X), by taking V (μ) = X ϕ(x) dμ(x), we have lim where we used Lebesgue's Dominated Convergence Theorem and the fact that γ (t) = v(γ (a)) = v(x) for all t ∈ I and η-a.e.(x, γ ) ∈ X × Y .By the arbitrariness of the sequence {t i } i∈N and using the Proposition 3.11, we conclude that lim Proof Without loss of generality we assume In particular, there exists η ∈ P p (X × Y ) with μ = e I η and supp η ⊆ graph S μ,G I .Let ε > 0 be fixed.There exist a natural number N ε > 0, positive constants λ 1 , . . ., λ Nε ∈ [0, 1], and a finite set {(x i , γ i )} i=1,...,Nε ⊆ graph S μ,G I such that, set we have W p (η, η ε ) ≤ ε/4, and so W p (μ t , μ ε t ) ≤ ε/4 by the 1-Lipschitz continuity of e t .

Constrained Mean Field Games Equilibria as Fixed Points of Random Lift
The concept of Mean Field Games (MFGs) was first introduced around 2006 by two independent groups, P. E. Caines, M. Huang, and R. P. Malhamé [13,14], and J.-M.Lasry and P.-L.Lions [17][18][19], motivated by problems in economics and engineering and building upon previous works on games with infinitely many agents.Roughly speaking, MFGs are game models with a continuum of indistinguishable, rational agents influenced only by the average behavior of other agents, and the typical goal of their analysis is to characterize their equilibria.In general, the equilibria are defined through a system of PDEs, known as MFG system, involving two unknown functions: the value function of the optimal control problem that a typical agent seeks to solve, and the timedependent density of the population of agents.
This interpretation fails in presence of state constraints, i.e., when agents are in the closure of a bounded domain.This case has been studied in [6][7][8], where the authors used a different approach to attack the problem.They define the equilibrium for the constrained problem replacing probability measures with measures on arcs, the so-called Lagrangian Formulation.
In this section, we want to show that the constrained MFG equilibria (defined in Definition 3.1 of [6]) are fixed-points of the set valued map in terms of random lift.We will extend also the notion given in [6] to the cases in which, besides the state constraints, there are also constraints in the dynamics of the agents, proving the existence of constrained MFG equilibria in this setting.
Let I = [a, b] ⊆ R be a compact interval, X be a finite-dimensional real space, ⊆ X be open and bounded.
We consider a set valued map F : I × P(X) × X ⇒ X expressing the microscopical dynamics of the agents as in (3.1), the cost for each agent, provided that the macroscopical evolution is described by θ ∈ P(C 0 (I ; X)), is given by a current cost : I × P(X) × X × X → [0, +∞], and an exit cost g : P(X) × X → [0, +∞[ as follows ) where I C is the indicator function defined as I C (v) = 0 if v ∈ C and +∞ otherwise, êt (γ ) = γ (t) for every γ ∈ C 0 (I ; X), λ ∈ {0, 1}, and we use the convention that λ • +∞ = 0 if λ = 0.
Namely, when the macroscopical evolution is described by θ ∈ P(C 0 (I ; X)), to have a finite cost each agent must respect the state constraint and stay in and, if λ = 1, he/she is allowed to move only according the trajectories of the differential inclusion.
Finally, we define A I : P p (X) ⇒ Z to be the set valued map given by i.e., A I (μ) is the set of fixed points of θ → ϒ J,θ I (μ).Given θ ∈ A I (μ), let η ∈ J,θ I (μ) be such that E p (η) θ .By disintegrating η w.r.t. the Borel map êa , we have η = μ ⊗ η x , where {η x } x∈X is a Borel family of probability measures, uniquely defined μ-a.e. and for μ-a.e.x ∈ X and η x -a.e.(z, γ ) ∈ X × Y , we have (z, γ ) ∈ graph S J,θ I with z = x, thus γ (a) = x and γ is a minimizer of J θ (•) among all ξ(•) ∈ AC(I ; X) satisfying ξ(a) = x.Since θ = pr 2 η, we conclude that for θ -a.e.γ ∈ Y we have that γ is a minimizer of J θ among all the absolutely continuous curves sharing its initial condition.
Proof First of all, using Lemma 3.3 in [6], Lemma 3.4 in [6] and Lemma 3.5 (if λ = 1) we have that the set valued map S J,θ I has closed graph and nonempty images for all θ ∈ P(Y ) supported on curves contained in .We prove that the images S J,θ I (x) are actually compact.If λ = 1 this follows immediately from the compactness of the trajectories of the differential inclusion.In the case λ = 0, we notice that all the minimizers γ ∈ S J,θ I (x) are 1 2 -Hölder continuous of constant M > 0. Indeed, for all γ ∈ S J,θ I (x) we have by hypothesis (L3) where γ (t) = x for all t ∈ I and γ ∈ Y .This yields γ 2 L 2 ≤ M, where M satisfies and c 0 , c 1 are the constants appearing in (L3).Therefore, we have that where R is the minimum radius of the ball B of center 0 that contains ¯ and the norm γ C 0,1/2 is defined as By Ascoli-Arzelà's Theorem, K is a compact set in C 0 (I ; R d ), and so S J,θ I (x) is compact.By e.g.Proposition 1.4.8 in [3] (θ , x) → S J,θ I (x) is upper semicontinuous with nonempty compact images.Since E p is the projection on the second component, the assumptions of Theorem 2.9 are satisfied.Therefore, for any μ ∈ P(X) there exists at least one θ ∈ P(Y ) such that θ ∈ A I (μ).

Cauchy-Lipschitz Theory for Continuity Equations
In the paper [5] the authors developed a powerful theory concerning optimal control problems in Wasserstein space.The main difference with the framework presented in Sect.3, where the point of view was more focused on the microscopic agent dynamics, this theory aims to directly deal with the controlled continuity equation, and associated control problem, from a purely Eulerian point of view.
In what follows, P c (R d ) will denote the set of compactly supported measures in P(R d ).
Definition 5.1 (Bonnet-Frankowska) Given T > 0 and a set-valued map V : [0, T ] × if there exists a measurable selection t → v t of V such that the trajectory-selection pair t → (μ t , v t ) solves the continuity equation ∂ t μ t +div(v t μ t ) = 0 in the sense of distributions.
The set of assumptions considered are the following: Hypotheses (DI).Let p ≥ 1 be given.For every R > 0, assume that the following holds with (iii) There exists a map l K (•) ∈ L 1 ([0, T ], R + ) such that for L 1 -almost every t ∈ [0, T ], any μ ∈ P(K) and every v ∈ V (t, μ), it holds Lip(v(•); K) ≤ l K (t).(iv) There a map L K (•) ∈ L 1 ([0, T ], R + ) such that for L 1 -almost every t ∈ [0, T ] and any μ, ν ∈ P(K), it holds According to the superposition principle (see e.g.Theorem 8.2.1 in [1]), given a solution of the Wasserstein differential inclusion (5.1) under the above assumptions there exists a measure η ∈ P(R d × I ) such that μ t = e t η and for η-a.e.(x, γ ) it holds γ (t) = v t (γ (t)) a.e. in [0, T ], γ (0) = x, and v t ∈ V (t, μ t ) for a.e.t ∈ [0, T ].Defined the set-valued map we notice that the solutions of (5.1) are in general a proper subset of the trajectories defined in Sect.3.Those trajectory turns to be more regular, and in [5] many deep results are proved about them.
Indeed, the regularity assumptions of Definition 5.1 and of the assumptions in (DI) implies that the microscopic trajectories are actually of class C 1,1 ([0, T ]) thus more regular than absolutely continuous or Lipschitz continuous.Furthermore, the characteristics form a global Lipschitz flow.From a microscopic model point of view, this implies that the agents are subject to an hidden interaction, forcing two agents sufficiently close each to the other to choose similar velocities.Furthermore, being the vector field v(•) locally Lipschitz continuous, during the evolution the mass cannot be splitted or concentrated to produce singular measures.We refer to [5] for other details about the comparison between those concepts of solutions.
Beside these differences, the random lift formalism allows to treat this concept of solution exactly as the previous ones.
In particular, we define Z = C 0 ([0, T ]; the open ball of radius r, i.e., B X (x, r) := {y ∈ X : d X (x, y) < r} (when there is no ambiguity, we simply write B(x, r)).The Lipschitz constant of a function f will be denoted by Lip(f ).
Given complete separable metric spaces (X, d X ), (Y, d Y ), we denote by P(X) the set of probability measures on X endowed with the weak * topology induced by the duality with the Banach space C 0 b (X) of real-valued continuous bounded functions.For any p ≥ 1, we set When X is a normed space, we define the p-moment of μ ∈ P(X) by m p (μ) := X x p X dμ(x).Given K ⊆ P(X), we say that K is tight if for all ε > 0 there exists a compact set K ε ⊂ X such that μ(X \ K ε ) ≤ ε for al μ ∈ K .For any Borel map r : X → Y and μ ∈ P(X), we define the push forward measure r μ ∈ P(Y ) by setting r μ(B) = μ(r −1 (B)) for any Borel set B of Y .Given finitely many complete separable metric spaces (X i , d X i ), i = 1, . . ., N, and any norm This metric gives to X the structure of a complete separable metric space.We denote the i-th projection pr i : X → X i by pr i (x 1 , . . ., x N ) = x i for all (x 1 , . . ., x N ) ∈ X, i = 1, . . ., N.
Given two sets X, Y , a set valued map (or multifunction) F (•) from X to Y (denoted by F : X ⇒ Y ) is a map F : X → P(Y ), where P(Y ) denotes the power set of Y , i.e., associating to each x ∈ X a (possible empty) subset F (x) ⊆ Y .The graph of a set valued map F is graph F : is both lower and upper semicontinuous at x.When Y is a linear space, say that F is convex valued (or that it has convex images) if F (x) is convex for every x ∈ X.When X is a measure space and Y is a topological space, we say that F is measurable if When X, Y are metric spaces, we say that is Lipschitz continuous if there exists L ≥ 0 (called a Lipschitz constant for F ) such that for all , where the sum and the product of sets are in the Minkowski sense: A + B = {a + b : a ∈ A, b ∈ B} and λA = {λa : a ∈ A} for every A, B ⊆ Y .We define Lip (F ) as the infimum of the constants L such that the above inequalty holds true.F has at most linear growth if there exists C > 0, x ∈ X and ȳ ∈ F ( x) such that d Y (y, ȳ) ≤ C(d X (x, x) + 1) for all x ∈ X, y ∈ F (x).
Conversely, assume that has closed graph, and let {(z n , x n , y n )} n∈N ∈ graph S be a sequence converging to (z, x, y) ∈ Z × X × Y .Choosing μ n = δ xn and η n = δ xn ⊗ δ yn , we have that {(z n , μ n , η n )} n∈N ∈ graph , which is closed.The result easily follows.
(5) Since K is relatively compact, by Prokhorov's Theorem (see e.g.Theorem 5.1.3in [1]) it is tight, i.e. for every ε > 0 there exists a compact set K ε ⊆ X such that μ(X \ K ε ) ≤ ε for all μ ∈ K .By (2), for all η ∈ z (K ), set μ η := pr 1 η ∈ K , it holds Since by assumption graph S |Kε is compact in X × Y , we have that z (K ) is tight, and so it is relatively compact in P(X × Y ), again by Prokhorov's Theorem.
By density, for every v ∈ X there exists a subsequence {v j k } k∈N such that v j k → v, and, by taking the limit of the above inequality using the continuity of γ (s), • and σ F (s,θ ,γ (s)) (•), we get γ (s), v ≤ σ F (s,θ ,γ (s)) (v), for all v ∈ X.By (F1), this implies γ (s) ∈ F (s, θ, γ (s)) for all s ∈ T , see e.g.page 59 of [2].Finally, by passing to the limit in x n = γ n (a) we have x = γ (a).Thus γ (•) ∈ S F,θ I (x), which proves the closedness of the graph.
We prove the assertion on compactness.We take two compact subsets K ⊆ Z, A ⊆ X, and let {γ n } n ⊂ S F I (K × A).Then there exist {θ (n) } n∈N ⊆ K, {x n } n∈N ⊆ A, C > 0 such that sup

Declarations
Competing Interests The authors declare no competing interests.
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