Mean-field optimal control in a multi-agent interaction model for prevention of maritime crime

We study a multi-agent system for the modeling maritime crime. The model involves three interacting populations of ships: commercial ships, pirate ships, and coast guard ships. Commercial ships follow commercial routes, are subject to traffic congestion, and are repelled by pirate ships. Pirate ships travel stochastically, are attracted by commercial ships and repelled by coast guard ships. Coast guard ships are controlled. We prove well-posedness of the model and existence of optimal controls that minimize dangerous contacts. Then we study, in a two-step procedure, the mean-field limit as the number of commercial ships and pirate ships is large, deriving a mean-field PDE/PDE/ODE model. Via Γ-convergence, we study the limit of the corresponding optimal control problems.


Introduction
Systems featuring interactions among multi-agents have attracted much attention of the scientific community in recent years as they find applications in various fields. They are a proper tool to study, e.g., biological aggregation as in flocks, swarms, or fish schools [19,14,37], crowd dynamics [2], emergent economic behaviors [16,21], consensus in collective decision-making [13,29], coordination and cooperation in robotics [17,34]. In this framework, Mathematical Analysis has played a role in the proof of well-posedness of the models, in the derivation of mean-field limit, and in the analysis of optimal control problems for this kind of models [25,12,4,3,30,7,6,1,5,24].
In this paper, we exploit the tools developed for the analysis of multi-agent systems to study optimal control in a model for the prediction of maritime crime. The majority of world's goods is carried by sea [22], but freedom of navigation is affected by the presence of modern maritime piracy, which poses serious threats to international traffic and individual safety. It is a priority to prevent crimes and suppress them [23].
To face this problem, we devise a model featuring three populations of agents, representing the type of ships. Our model is inspired by the macroscopic model (i.e., with a large number of ships) introduced in [18], but it differs from it in that our derivation starts from a microscopic model (i.e., with a finite number of ships). We briefly outline it in this introduction, referring to Section 3 for the precise description of all the features and assumptions on the model. We consider three populations: N commercial ships with trajectories X 1 , . . . , X N , M pirate (criminal) ships with trajectories Y 1 , . . . , Y M , and L coast guard (patrol) ships with trajectories Z 1 , . . . , Z L . The trajectory of each ship evolves in a time interval [0, T ] according to a specific dynamical law based on its type and on the presence of other surrounding ships, as we illustrate now.
Commercial ships tend to follow commercial routes, but their motion is affected by traffic congestion: a commercial ship obstructed by a high density of commercial ships travels slower than one with free space. Moreover, in presence of pirate ships, commercial ships are repelled by them and adjust their trajectory to travel far from danger. Hence, the n-th commercial ship evolves according to dX n dt (t) = v N n (X(t)) r(X n (t)) + 1) where v N n is a suitable function depending on all the other commercial ships needed for the congestion phenomenon, r is the vector field indicating the commercial route, and K cp is term due to the repulsion from pirate ships that adjusts the direction of the trajectory.
Pirate ships are attracted by commercial ships and are repelled by coast guard ships. Moreover, in absence of other ships, they travel randomly in search of targets. Hence, the m-th pirate ship evolves stochastically, according to where K pg and K pc are the repulsion and attraction terms with coast guard ships and commercial ships, respectively. The term (W m (t)) t∈[0,T ] is a Brownian motion accounting for the stochastic behavior mentioned above. Its effect is a white noise with coefficient √ 2κ added to the velocity of Y m .
Finally, for coast guard ships we only impose that are repelled one by each other and that their trajectory is controllable, at a cost. Hence, the ℓ -th coast guard ship evolves according to dZ ℓ dt where K gg is the repulsion term among coast guard ships and the u ℓ 's are the control. The search of coast guard ships for dangerous contacts between commercial and pirate ships will be driven by the optimal control of the system, based on the cost defined as follows. The cost of a control u = (u 1 , . . . , u L ) takes into account the effort in modifying the trajectories of coast guard ships (it can be thought as the cost of fuel), and the total number of dangerous contacts among commercial and pirate ships where H d is a compactly supported convolution kernel used for counting dangerous contacts and E denotes the expected value. We study the problem of finding a control that minimizes J N,M . In Section 4 we prove well-posedness of the model that describes the evolution and we prove existence of an optimal control.
Next, we proceed with the derivation of the mean-field limit of the optimal control problem. We carry out this analysis in two steps: first, we let M → +∞ (large number of pirate ships), and then N → +∞ (large number of commercial ships). The reason thereof is that the limit as M → +∞ is interesting per se, as we explain forthwith.
Under suitable conditions, in Section 7 (see Theorem 7.1 and Proposition 7.2) we show that, as M → +∞, the mean-field behavior of pirate ships is described by a probability distributionμ p . The trajectories of commercial shipsX n in this mean-field model satisfy dX n dt (t) = v N n (X(t)) r(X n (t)) + K cp * μ p (t)(X n (t)) , (1.4) which corresponds to (1.1) with the trajectories of pirate ships replaced by their mean-field behavior. The probability distributionμ p of pirate ships solves the diffusive PDE K pc (· −X n (t)) μ p = 0 . (1.5) This mean-field model is interesting per se when the precise location of pirate ships is not known, but one can only predict the probability of finding them in certain regions of the sea. Proving convergence of solutions of the original model to the mean-field model as M → +∞ requires some technical steps, mainly done following the guidelines in [9]. First, in Section 5 we introduce an auxiliary averaged model where the evolution of pirate ships is replaced by a single stochastic process evolving according to the same dynamics of (1.2), i.e., whereX n evolves according to (1.4),μ p being the law of the stochastic process (Ȳ (t)) t∈[0,T ] . In Section 5 we prove well-posedness for this averaged model, using a fixed-point argument.
Solutions to the original model converge, as M → +∞, to solutions of this auxiliary averaged model. To see this, in Propositions 6.1-6.2 we rely on a propagation of chaos principle [15], from which we deduce that solutions to (1.2) are independent and identically distributed stochastic processes, if so are the initial conditions. Then, a Glivenko-Cantelli-type result allows us to deduce convergence of the empirical measures of the Y m 's to their common lawμ p . The parabolic PDE (1. As a consequence, optimal controls for the original problem converge as M → +∞ to optimal controls for the limit problem, see Proposition 7.4. This concludes the analysis as M → +∞. The next step is to study the mean-field limit as the number of commercial ship is large, i.e., when N → +∞. In Theorem 8.1 and Proposition 8.3, we show that the mean-field limit of commercial ships is described in terms of their distribution µ c , which solves a scalar conservation law with a nonlocal flux, apt to describe traffic flow in sea. More precisely, µ c is a solution to the PDE ∂ t µ c + div x v η * 2 µ c r + K cp * µ p µ c = 0 , where v η * 2 µ c arises from the limit of the congestion velocities and µ p is the probability distribution of pirate ships, evolving according to the parabolic PDE ∂ t µ p − κ∆ y µ p + div y 1 L L ℓ=1 K pg (· − Z ℓ (t)) − K pc * µ c µ p = 0 .
Under suitable assumptions, in Theorem 8.4 we prove uniqueness of solutions to this PDE system and, as observed in Remark 8.5, that the measures are absolutely continuous, i.e., µ c = ρ c dx and µ p = ρ p dy .
We conclude the paper by finding in Theorem 8.6 the Γ-limit of the costs J N defined in (1.6) as N → +∞. It is given by the cost for the latter mean-field system Also in this case, we deduce convergence of optimal controls as N → +∞, see Proposition 8.7. The limit problem is an optimal control problem with a finite number of coast guard ships driving the densities of commercial and criminal ships.

Notation and preliminary results
2.1. Basic notation and preliminary results. Given a matrix A, we let |A| its Frobenius norm. We shall often consider matrices of the form A ∈ R 2×d . By writing A = (A 1 , . . . , A d ) we make explicit its columns A i ∈ R 2 .
Throughout the paper, we shall systematically apply Grönwall's inequality. We recall that if u, v, w : [0, T ] → R are continuous and nonnegative functions satisfying If not specified otherwise, we let C denote a constant that might change from line to line. We make precise the dependence of C on other constants when it is relevant for the discussion.

Stochastic processes and Brownian motion.
For the theory of stochastic processes and stochastic differential equations we refer to the monographs [27,28,31]. Here we recall some basic facts and definitions used in the paper.
We fix a probability space (Ω, F , P) used throughout the paper. By a.s. (almost surely) we mean P-almost everywhere. We let E denote the expectation.
A filtration on (Ω, F , P) is a collection of σ -algebras (F t ) t∈[0,T ] increasing in t, i.e., F s ⊂ F t for s ≤ t. When (Ω, F , P) is a complete probability space, (F t ) t∈[0,T ] is said to satisfy the usual conditions if it is right-continuous (i.e., F s = t>s F t for all s) and if N P ⊂ F 0 , where N P = {A ⊂ Ω s.t. A ⊂ B with B ∈ F and P(B) = 0} (if (Ω, F , P) is complete, this means that F 0 contains P-null sets).
A stochastic process is a parametrized collection of random variables (S(t)) t∈[0,T ] defined on (Ω, F , P) and assuming values in R d (equipped with σ -algebra of Borel sets). Given t ∈ [0, T ] and ω ∈ Ω, we will write S(t, ω) = S(t)(ω) for the realization of the random variable S(t) at ω . A path of the stochastic process is a curve in R d obtained as the realization t → S(t, ω) for some ω ∈ Ω. A stochastic process (S(t)) t∈[0,T ] is adapted to a filtration Let (F t ) t∈[0,T ] be a filtration. A d-dimensional Brownian motion (or Wiener process) is a R d -valued stochastic process (W (t)) t∈[0,T ] , adapted to (F t ) t∈[0,T ] , a.s. with continuous paths such that: 1 Equivalently, it has components W (t) = (W 1 (t), . . . , W d (t)) with (W 1 (t)) t∈[0,T ] , . . . , (W d (t)) t∈[0,T ] independent 1-dimensional Brownian motions. 1 One can speak of a Brownian motion without introducing the filtration (Ft) t∈[0,T ] by replacing the condition that W (t) − W (s) is independent of Fs with the requirement that it has independent increments.

Stochastic Differential Equation.
For the general theory about SDEs, we refer to [27,28,31]. We recall here some basic facts. Let (F t ) t∈[0,T ] be a filtration satisfying the usual conditions, let (W (t)) t∈[0,T ] be a d-dimensional Brownian motion, and let us consider an initial datum S 0 given by a F 0 -measurable random variable. 2 However, in this paper we are only interested to a specific class of SDEs, i.e., those with a constant dispersion matrix of the form A stochastic process (S(t)) t∈[0,T ] is a strong solution to (2.1) if (S(t)) t∈[0,T ] has a.s. continuous paths, it is adapted to the filtration (F t ) t∈[0,T ] , satisfies a.s.
T 0 |b(t, S(t))|dt < ∞ and for every t ∈ [0, T ] For this class of SDEs, it is well-known that the well-posedness theory is simpler [27,Equation (2.34)] and requires weaker assumptions on the initial datum S 0 than those usually stated in general theorems. For the reader's convenience we state and prove the result in the form needed in this paper, as we did not find a precise reference in the literature. Besides, some of the tools used in the proof will be exploited later in the paper. The result is stated with the Euclidean norm | · | on R d , but we remark that it holds true when replacing it with any equivalent norm, e.g., also max h |S h |, as long as the assumptions on b are satisfied with that norm.
Proof. The scheme of the proof is the classical one, see [28,Theorem 3.3]. Let us fix ω ∈ Ω such that |S 0 (ω)| < +∞ and t → W (t, ω) is continuous, which occurs almost surely. We consider the Picard iterations Note that the curve t → S j (t, ω) is continuous. First of all, let us prove that for all j and for all t ∈ [0, T ]

4)
In this case, one implicitly considers a filtration constructed from (W (t)) t∈[0,T ] by letting F W t be σalgebra generated by {W (s) | s ≤ t} . If the filtration needs to satisfy the usual conditions, F W t is modified with the augmentation Ft defined as the σ -algebra generated by F W t and N P , see [28, p. 16] which proves (2.4). In particular, Thanks to this, we show that for a suitable constant C(ω) depending on ω . Indeed, for j = 0 , by the linear growth of b we have that for every s ∈ [0, T ] Moreover, by the local Lipschitz continuity of b we have that for every s ∈ [0, T ] Assuming (2.6) true for j − 1 , we have that This implies that S j (·, ω) is a Cauchy sequence in the uniform norm, since for j ≥ i Thus there exists a continuous curve S(·, ω) such that We have constructed S(·, ω) for a.e. ω ∈ Ω. The stochastic processes ( S j (t)) t∈[0,T ] are adapted to the filtration (F t ) t∈[0,T ] and have a.s. continuous paths. This implies that the limit (S(t)) t∈[0,T ] is a stochastic process adapted to the filtration (F t ) t∈[0,T ] and has a.s. continuos paths. Moreover, passing to the limit in (2.2) for a.e. ω ∈ Ω, it is a strong solution to (2.1).
Uniqueness is proven in a more general setting in [27, Theorem 2.5] via a stopping time argument.
where the infimum is taken over all transport plans γ ∈ P(B×B) with marginals π 1 # γ = µ 1 and π 2 # γ = µ 2 , where π i is the projection on the i -th component. We shall often exploit the dual formulation of the 1 -Wasserstein distance. By Kantorovich's duality, [39,Theorem 5.10] we have that Since d is a distance on a metric space, a d-convex function ψ is a Lipschitz function with Lipschitz constant 1 and it coincides with its d-transform, cf. [39,Particular Case 5.4]. Hence, if ψ is a Lipschitz function with Lipschitz constant Lip(ψ), we have that When in this paper we refer to Kantorovich's duality, we apply this inequality. Note that the condition ψ ∈ L 1 (µ 1 ) ∩ L 1 (µ 2 ) is satisfied since |ψ(x)| ≤ |ψ(0)| + Lip(ψ)|x| and µ 1 , µ 2 ∈ P 1 (B).

Wiener space.
Given an interval [0, T ], we shall consider the so-called Wiener space of R d -valued continuous functions C 0 ([0, T ]; R d ), equipped with the uniform norm. Given t ∈ [0, T ], we consider the evaluation function ev t : C 0 ([0, T ]; R d ) → R d defined by ev t (ϕ) := ϕ(t) for every ϕ ∈ C 0 ([0, T ]; R d ). The family of evaluation functions {ev t } t∈[0,T ] generates a σ -algebra on C 0 ([0, T ]; R d ), which coincides with the Borel σ -algebra with respect to the uniform norm in C 0 ([0, T ]; R d ). 3 This is generated by cylindrical sets of the form Let (S(t)) t∈[0,T ] be a R d -valued stochastic process a.s. with continuous paths. This means that there exists an event E ∈ F such that P(E) = 1 and t → S(t, ω) is continuous for all ω ∈ E . We can redefine S(t, ω) = 0 for all t ∈ [0, T ] when ω ∈ Ω \ E . This new stochastic process is indistinguishable from the previous one and satisfies S(·, ω) ∈ C 0 ([0, T ]; R d ) for all ω ∈ Ω. The stochastic process (S(t)) t∈[0,T ] can be regarded as the random variable The σ -algebra generated by this random variable is the σ -algebra generated by sets of the form For these sets we have that and thus the σ -algebra generated by S : Ω → C 0 ([0, T ]; R d ) coincides with the σ -algebra generated by the family {S(t)} t∈[0,T ] of random variables S(t, ·) : Ω → R d , i.e., the σalgebra generated by the stochastic process.

Description of the model
To better describe the phenomena that we aim to capture, we introduce all the ingredients that enter in the model step by step. For the reader's convenience, all the unknowns, the parameters, and the initial data of the model are summarized in Tables 1-5.
The model is an evolutionary system, analyzed in a fixed time interval [0, T ].

Ships.
The system describes the evolution of N commercial ships, M pirate (criminal) ships, and L coast guard (patrol) ships, whose trajectories are curves X n : , . . . , L} , respectively. We shall often collect the trajectories based on their type by considering the matrixvalued curves X = ( T ] → R 2×L . The letters X , Y , Z will unambiguously indicate the type of ship, even when decorated, e.g., asX , X , or with superscripts and subscripts.
Hereafter, whenever a variable is related to commercial, pirate, or guard ships, it is be indexed with the superscript c, p, or g , respectively.

Item
Meaning Comment  Evolution of commercial ships.
Step 1. We start by describing the evolution of commercial ships in safe waters (absence of pirate ships) and in absence of congestion in the traffic. We assume that there is a vector field r : R 2 → R 2 indicating safe commercial routes. In this ideal setting, commercial ships evolve according to the ODEs    dX n dt (t) = r(X n (t)) , is the initial position of commercial ships. We shall assume that r is globally Lipschitz continuos.
Step 2. To include congestion in the model, n weighs the speed of the trajectory of the n-th commercial ship according to the presence of all the other commercial ships:    dX n dt (t) = v N n (X(t))r(X n (t)) , X n (0) = X 0 n , n = 1, . . . , N . The assumptions on v N needed throughout the paper are the following: v N is Lipschitz continuous with respect to the max norm with a Lipschitz constant independent of N , i.e., For v N we have in mind a precise expression, that will be used in Section 8. We consider a globally Lipschitz smooth convolution kernel η : R 2 ×R 2 → [0, 1] satisfying η(X, 0) = 0 .

The quantity
suitably counts 6 the number of commercial ships around the n-th commercial ship at time t. Hence, the quantity can be regarded as the density of commercial ships around the n-th commercial ship at time t. The precise expression of the scaling factor 1 N −1 is relevant only to interpret the previous expression as a density and can, in fact, be replaced by a sequence converging to zero with the same rate of 1 N . Given a Lipschitz function v : [0, 1] → [0, v max ], the corrected speed of the n-th commercial ship depends on the density of its surrounding ships as follows: To model congestion, v must be assumed to be non-increasing in the density.
Step 3. Eventually, let us modify the dynamics of commercial ships in presence pirate ships. We consider a globally Lipschitz vector-valued interaction kernel K cp : R 2 → R 2 (here cp stands for commercial-pirate). To model repulsion of the n-th commercial ship from the pirate ships, we modify the direction of the trajectory X n (t) by averaging the vectors X n (0) = X 0 n , n = 1, . . . , N . For K cp we have in mind the following expression where H cp has compact support with a radius given by the length for which the presence of a pirate ship at Y m (t) affects the trajectory X n (t). An example of H cp is H cp (w) = h(|w|) |w| , where h is compactly supported in (0, +∞), so that |Xn(t)−Ym(t)| and Xn(t)−Ym(t) |Xn(t)−Ym(t)| is, for X n (t), the direction pointing opposite to Y m (t).  Table 2. Summary of functions used in the model for evolution of commercial ships. 6 For example, letη ∈ C ∞ c (R 2 ) be supported in a ball B 2δ of radius 2δ withη = 1 on B δ . If η(X, X ′ ) =η(X − X ′ ) , then N n ′ =1η (Xn(t) − X n ′ (t)) (approximately) counts the number of ships in a δ -neighborhood of Xn(t) (around all directions). Instead, If η(X, X ′ ) =η(X − X ′ − δr(X)) , then N n ′ =1η (Xn(t) − X n ′ (t) − δr(Xn(t))) (approximately) counts the number of commercial ships obstructing the commercial route in front of Xn(t) .
Evolution of pirate ships. Step 1. Pirate ships are are repelled by guard ships and are attracted by commercial ships. To model this, we consider globally Lipschitz vector-valued interaction kernels K pg : R 2 → R 2 and K pc : is the initial position of pirate ships. For the precise form of K pg , K pc , see the analogous discussion for commercial ships done after (3.1).
Step 2. In absence of commercial and guard ships, pirate ships explore the environment in search of targets by navigating randomly. To model this, we add a stochastic term in the evolution of pirate ships, by considering M Brownian motions (W 1 (t)) t∈[0,T ] , . . . , (W M (t)) t∈[0,T ] . The pirate ships then evolve according to the following SDEs Evolution of guard ships. The last part of the system describes guard ships. In absence of other ships, guard ships tend to repel each other. To model this, we consider globally Lipschitz vector-valued interaction kernel K gg : R 2 → R 2 . In this setting, the guard ships evolve according to where Z 0 = (Z 0 1 , . . . , Z 0 L ) ∈ R 2×L is the initial position of guard ships. We do not require more on the dynamics of guard ships, as we want the global dynamics of the system to be governed by the optimal control policy for guard ships.  Table 3. Interaction kernels used in the model.

Controls.
We consider a set of admissible controls U ⊂ R 2×L . We assume U to be compact. A fixed control u = (u 1 , . . . , u L ) ∈ L ∞ ([0, T ]; U) drives the evolution of guard ships as follows: Full model. In conclusion we are interested in the following ODE/SDE/ODE model: (3.2) (The first equation is expressed as an SDE to stress that the solution X is a stochastic process. However, given a trajectory Y , the first equation is, in fact, and ODE.) We prove well-posedness for (3.2) in Subsection 4.1.
Initial data. The initial data in (3.2) will be given by

Item
Meaning Comment  Table 4. Summary of initial data.
Optimal control. As previously mentioned, the dynamics of guard ships will be driven by an optimal control. To define the cost, we consider a bounded and globally Lipschitz function is close to X n (t) and is small when Y m (t) is far from X n (t) (e.g., when H d is compactly supported), this function can be used to count contacts between commercial and pirate ships (the superscript d stands for "danger"). Hence we consider the cost functional where the stochastic processes (X(t)) t∈[0,T ] = (X 1 (t), . . . , X N (t))) t∈   We remark that the solutions depend on N and M . Not to overburden the notation, in this section we drop the dependence on N and M , as we will not consider limits as N → +∞ or M → +∞.
Proposition 4.1. Assume the following: Then there exists a unique strong solution to (3 Proof. We start by noticing that the ODEs involving the variables Z ℓ are decoupled from the equations involving X n and Y m . Given a control u = (u 1 , . . . , . . , L . We observe that there exists a unique solution for all times t ∈ [0, T ] to the previous ODE system. Too see this, we introduce the function and we notice that the system reads , where Z = (Z 1 , . . . , Z L ). The right-hand side f (t, Z) is a Carathéodory function, globally Lipschitz-continuous in the Z variable (with Lipschitz constant independent of t). These properties are sufficient for the well-posedness of the ODE. 7 We remark that solutions to (4.1) are bounded. Indeed, where the constant C depends on K gg and U (compact). We exploit the solution Z(t) to solve the ODE/SDE/ODE system, which now we write in a more compact way. Let us introduce the R 2×(M+N ) -valued stochastic process ( in the first block for consistency later). We con- we adopt the short-hand notation σW to denote the element in R 2×(M+N ) with columns (σW ) 1 , . . . , (σW ) M+N ∈ R 2 given by (σW . This is the reason why we chose to put the Y m 's in place of the X n 's in the first block of S . We are now left to check that the conditions for the existence and uniqueness stated in Proposition 2.1 are satisfied by (4.6). By the continuity of Z(t), the function t → b(t, S) is continuous for every S . Let i ∈ {1, . . . , M } , so that b i is given by (4.4). By the Lipschitz continuity of K pg , we have that The result is classical: one considers the Picard operator S : ds , which is a contraction with respect to the norm (equivalent to the uniform norm) |||ϕ||| α := sup t∈[0,T ] e −αt |ϕ(t)| for a suitable α > 0 (depending on the Lipschitz constant of f ).
Reasoning analogously for K pc , it follows that where we used the continuity, and thus boundedness, of Z ℓ (t) for t ∈ [0, T ]. Let us check the Lipschitz continuity condition. By the Lipschitz continuity of K pg and K pc , we have that where the constant C depends on K pg , K pc . (In fact, b i is even globally Lipschitz continuous for i ∈ {1, . . . , M } ). Let now i ∈ {M + 1, . . . , M + N } , so that b i is given by (4.5). By the boundedness of v N , by the bound r(x) ≤ C(1 + |x|), and reasoning for K cp as in (4.7), we have that To check the local Lipschitz-continuity of b i , let us fix R > 0 . For t ∈ [0, T ] and max h |S h | ≤ R , max h |S ′ h | ≤ R , by the boundedness and the Lipschitz property of v N (recall that it has a Lipschitz constant independent of N ), and by the Lipschitz continuity of r and K cp , we have that where the constant C depends on v N , r, and K cp (independent of N ). Choosing C R = C(1 + R) we get the desired inequality. Applying Proposition 2.1, we conclude the proof of existence and uniqueness. Moreover, we also get E(max h S h ∞ ) < +∞ and, in particular, E(max m Y m ∞ ) < +∞.

4.2.
Existence of an optimal control for the ODE/SDE/ODE model. Let J N,M be the cost defined in (3.3). We have the following result concerning existence of optimal controls.

Proposition 4.2. Under the assumptions of Proposition 4.1, there exists an optimal control
Proof. The result is obtained via the direct method in the Calculus of Variations. We divide the proof in steps for the sake of presentation.
. We claim that u * is an optimal control.
To prove the claim, let us fix ( , the strong solutions to (3.2) corresponding to the controls u j obtained in Proposition 4.1. We adopt the notation of the proof of Proposition 4.1 and let S = (Y 1 , . . . , Y M , X 1 , . . . , X N ). In this way, for every j we have that (we stress the dependence of f u j on the controls u j ) and (we stress the dependence of the drift vector R 2×L on the trajectories Z j ).
Step 2. (Identifying the limit of Z j ) We remark that (4.3) yields Z j ∞ ≤ C for every j , where C depends on Z 0 , T , K gg , and U . Let us check that the Z j 's are also equicontinuous. By (4.2), for every j and for s ≤ t we have that where C depends on Z 0 , T , K gg , and U (compact). By Arzelà-Ascoli's theorem we obtain This, together with the convergence u j * ⇀ u * and Step 3. (Identifying the limit of S j ) We let (S * (t)) t∈[0,T ] be the R 2 -valued stochastic process obtained as the strong solution to (4.12) We estimate the former integrand by exploiting the Lipschitz property of b Z j ,i obtained in (4.9) and (4.11) where the constant C depends on K pg , K pc , K cp , v N , and r (independent of N ). To estimate the latter integrand in (4.12), we resort to the definition of b Z . By (4.4), for i = 1, . . . , M we get that where the constant C depends on K pg . For i = M +1, . . . , M +N , by (4.5) we have instead that |b Z j ,i (r, S * (r)) − b Z * ,i (r, S * (r))| = 0 . We observe that Proposition 2.1 also gives us We remark that P k A k = 1 , since a.s. max h S * h ∞ < +∞. Let us fix ω ∈ A k and such that (4.12)-(4.14) hold true. Then we have that Integrating on A k , we get that By Grönwall's inequality we deduce that and, in particular, Step 4. (Limit of the cost) Let us show that Since u j is a minimizing sequence, this will be sufficient to conclude that J N,M (u * ) = min u J N,M (u). By sequential weak semicontinuity of the L 2 -norm we get that . Then, using the fact that H d is bounded, by the Dominated Convergence Theorem as j → +∞. By the superadditivity of the lim inf we conclude the proof.
5. An averaged ODE/SDE/ODE system 5.1. Introducing the averaged ODE/SDE/ODE system. To study the mean-field limit of (3.2) as M → +∞, we consider an averaged ODE/SDE/ODE system, where the trajectories Y m (t) are replaced by a single trajectoryȲ (t), interacting with the other agents via its probability distribution. More precisely, let (W (t)) t∈[0,T ] be a R 2 -valued Brownian motion and consider the problem We start by giving a precise definition for the notion of solutions to the previous system.

5.2.
Well-posedness of the averaged ODE/SDE/ODE system. Let us prove the following well-posedness result.
Proposition 5.2. Assume the following: Then there exists a unique strong solution to (5.1).
Proof. As recalled in the proof of Proposition 4.1, for every control hence Z ℓ (t) will be treated as fixed in the following.
The proof now mainly follows the lines of [9, Theorem 3.1]. For the sake of brevity, we Step 1. (Decoupling the system) Let us fix µ ∈ P 1 (C 0 ) (µ plays the role ofμ p in the equation and is used to apply a fixed-point argument). Let us consider the decoupled system X n (0) = X 0 n , n = 1, . . . , N , where the Z ℓ (t) are obtained in (5.2). Substep 1.1. We start by commenting about the existence (and uniqueness) of continuous curves (5.3). For this, we need to check the conditions for well-posedness of ODE systems. Let us consider the function g µ = (g µ,1 , . . . , g µ,N ) : for n = 1, . . . , N . The system then reads The dependence of g µ on the time variable t is only due to the terms which are continuous in t. This follows from, e.g., the Dominated Convergence Theorem by observing that the Lipschitz continuity of K cp yields The computations are analogous to those in (4.11), the only difference being in the term In conclusion, g µ (t, X) is continuous in t and locally Lipschitz in X with respect to the max norm. By Picard-Lindelhöf's theorem, the ODE system (5.6) admits a unique solution for small times. For existence for all times, with computation analogous to those in (4.10) we observe that we have linear growth for g µ , i.e., the constant C above depending on v N ∞ , r, and K cp . This upper bound allows for a Grönwall inequality. Indeed, and, in particular, boundedness of solutions in terms of the initial datum X 0 and final time T (in addition to v N ∞ , r, and K cp ). This is enough to deduce global existence in time. Substep 1.2. Given the continuous curves X and Z obtained previously, we consider the SDE (5.4). We rewrite this SDE by introducing the drift vector b X : and by considering the constant dispersion matrix σ = √ 2κ Id 2 , so that the SDE reads For the existence and uniqueness of a strong solution to this SDE, we check that the assumptions of Proposition 2.1 are satisfied. The drift b X is continuous in t: it follows from the continuity of the curves X n and Z ℓ . The drift b X is globally Lipschitz continuos in Y . Indeed, we have that the constant C only depending on the Lipschitz constants of K pg and K pc . Finally, b X satisfies the linear growth condition. This follows from (4.7) and the analogous condition for K pc , which yield where the constant C depends on K pg , K pc , Z ∞ , and max n X n ∞ and we used the boundedness of X obtained in (5.8).
We are in a position to apply Proposition 2.1, which also gives us that Step 2. (Fixed-point argument) Let us implement the machinery to carry out a fixed-point argument.
Substep 2.1. (Definition of Picard operator) We consider the functional L : be the unique solution to (5.3)-(5.4) obtained as explained in the previous step. Then we set L(µ) := Law( Y ), which belongs to P 1 (C 0 ) as explained in the previous step. We shall show that L is a contraction with respect to a suitable auxiliary distance on P 1 (C 0 ), to deduce the existence of a fixed point.
We start by observing that from (5.10), from the definition of b X in (5.9), by the Lipschitz continuity of K pc , and by the Lipschitz continuity of b X ′ obtained in (5.11), we have that a.s.
the constant C depending only on the Lipschitz constants of K pg and K pc .
The curves X and X ′ are solutions to (5.6). As obtained in (5.8), they are bounded by a constant R > 0 depending on the initial datum X 0 , the final time T , and parameters of the problem ( v ∞ , r, and K cp ), i.e., max n X n ∞ ≤ R , max n X ′ n ∞ ≤ R . We recall that g µ and g µ ′ are locally Lipschitz, hence there exists C > 0 (depending on R ) such that (5.7) is satisfied. It follows that for Let us now apply the definition of g µ and g µ ′ in (5.5) to estimate for n = 1, . . . , N and r ∈ [0, s] |g µ,n (r, X ′ (r)) − g µ ′ ,n (r, where the constant C depends on v ∞ . We observe that by the Lipschitz continuity of x → K cp ( X ′ n (r) − x) and by Kantorovich's duality, where C is the Lipschitz constant of K cp . To bound this term, let us fix an optimal plan γ ∈ P(C 0 ×C 0 ) with marginals π 1 # γ = µ, π 2 # γ = µ ′ and satisfying We remark that γ(r) = (ev r ) # γ ∈ P(R 2 ×R 2 ) has marginals π 1 # (ev r ) # γ = (ev r ) # π 1 # γ = µ(r) and π 2 # (ev r ) # γ = (ev r ) # π 2 # γ = µ ′ (r), hence, by optimality of W 1 and by the definition of ||| · ||| α in (5.13), we obtain for r ∈ [0, s] Integrating in r , we get that Putting together (5.16)-(5.19) we conclude that By Grönwall's inequality we conclude that (5.20) To sum up, the constant C in the previous formula depends on: X 0 , T , v ∞ , r, and K cp .
Multiplying both sides by e −αs and using that e −αs ≤ e −αr we get that a.s. for s ∈ [0, t] Taking the supremum for s ∈ [0, t] and the expectation, we deduce that and thus, by Grönwall's inequality, Keeping track of the constant C , it depends on: X 0 , T , v ∞ , r, K cp , K pg , and K pc . Substep 2.7. (Choice of α and end of proof of contraction property) We choose α > 0 in such a way that where C is the constant obtained in (5.21). In this way, by (5.14) and (5.21) we conclude that W 1,α (L(µ), L(µ ′ )) ≤ C α W 1,α (µ, µ ′ ) , i.e., L : P 1 (C 0 ) → P 1 (C 0 ) is a contraction with respect to the equivalent Wasserstein distance W 1,α . As such, it admits a unique fixed pointμ p ∈ P 1 (C 0 ).
Step 3. Given the fixed pointμ p ∈ P 1 (C 0 ) of L, we defineX = (X 1 , . . . ,X N ) as the solution to (5.3) corresponding toμ p , and then we letȲ be the solution to (5.4) corresponding toX . Sinceμ p is a fixed point, we have that L(μ p ) =μ p , i.e., Law(Ȳ ) =μ p . Hence we found the unique strong solution to the coupled system. This concludes the proof.
Remark 5.3. By (5.8), it follows that max n X n ∞ is bounded by a constant depending on the initial datum X 0 , the final time T , v N ∞ , r, and K cp .
6. Propagation of chaos Proposition 6.1. Assume the following: Proof. We fix Z = (Z 1 , . . . , Z L ) as the solution to . . , L , which is independent of m since it is decoupled from the first two sets of equations.
Let m ∈ {1, 2} . We resort to some tools already considered in Step 2 in the proof of Proposition 5.2. As in that proof, we set C 0 := C 0 ([0, T ]; R 2 ).
We observe that Law( Y 0 1 ) = Law( Y 0 2 ), as by (6.5) they coincide with the common law of the identically distributed random variables given by the initial data Y 0 1 , Y 0 2 . This is the base step of an induction argument. Let j ≥ 1 and assume Law( With this notation, for ω ∈ Ω such that W m (·, ω) is a continuous path (this occurs a.s.). Then we have that , that is our claim (6.4).
Step 2. (Exploiting the fixed point) For m = 1, 2 we consider the functionals L m = L Y 0 m ,Wm : P 1 (C 0 ) → P 1 (C 0 ) defined as in Step 2 in the proof of Proposition 5.2 (we stress here the dependence on m to keep track of the dependence on the initial datum Y 0 m and the Brownian motion W m ). Given µ ∈ P 1 (C 0 ), we let X = ( X 1 , . . . , X N ) and ( Y m (t)) t∈[0,T ] be the unique solution to the decoupled system (6.2)-(6.3). Then we set L m (µ) := Law( Y m ). By the discussion in Step 1 we have that L 1 (µ) = L 2 (µ).
The solutionX m = (X m,1 , . . . ,X m,N ) is then obtained as the solution to (6.2) corresponding toμ p . Thus it does not depend on m, yieldingX 1 =X 2 . Proposition 6.2. Assume the following: For every m = 1, . . . , M , letX = (X 1 , . . . ,X N ), (Ȳ m (t)) t∈[0,T ] , Z = (Z 1 , . . . , Z L ) be the unique strong solution to 9  Proof. The leading idea of the proof is to to writeȲ m in terms of the initial datum Y 0 m and the Brownian motion W m .
We consider the solution operator S : R 2 ×C 0 → C 0 defined by S(ξ, w) := ϕ, where ϕ is the unique solution to the integral equation The fact that there exists a unique solution to the previous problem follows from the fact that the operator Ψ : is such that Ψ(ξ, ·, w) : C 0 → C 0 is a contraction with respect to the auxiliary norm |||ϕ||| α := sup t∈[0,T ] (e −αt |ϕ(t)|) for a suitable α > 0 . Indeed, by the Lipschitz continuity of bX , and thus it has a unique fixed point.

Mean-field limit for a large number of pirate ships
In this section we study the limit of the problem as M → +∞. For this reason we will stress the dependence of initial data and solutions on M . Still, we do not stress dependence on N , not to overburden the notation. (The first placeholder is kept free for the time variable.) If max m E( S m ∞ ) < +∞ then a.s. ν M ∈ P 1 (C 0 ([0, T ]; R 2 )). Indeed, We set ν M (t, ω) := (ev t ) # ν M (·, ω) for all ω ∈ Ω and t ∈ [0, T ]. With a slight abuse of notation, we let ν M (t) denote the random measure ν M (t) : Ω → P(R 2 ).
In fact, by the boundedness of U , this is equivalent to requiring that u M ⇀ u weakly in L 1 ([0, T ]; U ) . 12 This corresponds to the original ODE/SDE/ODE system (3.2) with initial data X 0 , Y 0 , Z 0 and with control u M . The solution is provided by Proposition 4.1. We stressed the dependence on M since we are interested in the limit as M → +∞ . 13 Corresponding to the initial data X 0 ,Ȳ 0 , Z 0 , with Brownian motion W , and control u . We recall that the solution is provided by Proposition 5.2. 14 This corresponds to the averaged ODE/SDE/ODE system (5.1) with initial data X 0 , Y 0 m , Z 0 , with Brownian motion Wm , and control u . The solution is provided by Proposition 5.2. Note that we applied Proposition 6.1 to deduce that   Step 2.

Our first task is to prove that
) . Then, we exploit the properties of v N , r, and K cp and (7.7) to get from (7.2) that a.s. for 0 ≤ s ≤ t and n = 1, . . . , N To estimate the term involving |K cp * ν p M (r)(X M n (r)) − K cp * μ p (r)(X n (r))| in (7.8), we exploit Kantorovich's duality and the Lipschitz continuity of K cp to get that a.s.

(7.9)
We bound W 1 (ν p M (r),ν p M (r)) using for a.e. ω ∈ Ω as an admissible transport plan the diagonal transport γ(ω) = 1 N N n=1 δ (Y M n (r,ω),Ȳ M n (r,ω)) to obtain that a.s. (7.10) To estimate the term involving K cp * μ p (r)(X n (r)) in (7.8), we use the fact that |K cp (z)| ≤ |K cp (0)| + C|z| to get that where in the last inequality we used that which is finite, sinceμ p ∈ P 1 (C 0 ) by Proposition 5.2. By Remark 5.3 we recall that max n X n ∞ is bounded by a constant depending on max n X 0 n ∞ , T , v N ∞ , r, and K cp . Hence |K cp * μ p (r)(X n (r))| ≤ C . (7.11) Then we can proceed with the estimate in (7.8): By (7.9)-(7.11) and by exploiting also the Lipschitz continuity of r and v N , we obtain that Taking the supremum in s, the maximum in n, and then the expectation, we obtain that Step 3. (Grönwall's inequality) Putting together (7.6) and (7.12) we have that for every By Grönwall's inequality, we deduce that for every t ∈ [0, T ] In particular, where the constant depends additionally on T .
where the constant C depends on K gg . By Grönwall's inequality, it follows that where the finiteness of the last term follows from Proposition 5.2. We conclude that as M → +∞. This concludes the proof of (7.5).
thus, taking the expectation, we obtain in particular that Using the fact thatμ p (t) = Law(Ȳ (t)) and ξ(T, ·) ≡ 0 , we get that This concludes the proof.

7.2.
Limit of optimal control problems as M → +∞. Let us consider the following cost functional for the limit problem obtained in (7.1). Let J N : On the one hand, by definition (3.3), we have that where the stochastic processes ( T ] (and the curve Z M = (Z M 1 , . . . , Z M L )) are the unique strong solution to (7.2). On the other hand, we have that where the curveX = (X 1 , . . . ,X N ), the stochastic process (Ȳ (t)) t∈[0,T ] with lawμ p (and the curve Z = (Z 1 , . . . , Z L )) are the unique strong solution to (5.1). By the weak sequential lower semicontinuity of the L 2 -norm, we have that Let us prove the convergence 18) as M → +∞. This will conclude the proof of (7.17).
We exploit the equality to deduce that We estimate the first term on the right-hand side of (7.19) by using the fact that, by the Lipschitz continuity of H d , a.s. for every t ∈ [0, T ] We estimate the second term on the right-hand side of (7.19) by Kantorovich's duality, which by the Lipschitz continuity of H d (X n (t) − ·) yields a.s. for every t ∈ [0, T ] Putting together the previous inequalities, we conclude that where the curveX , the stochastic process (Ȳ (t)) t∈[0,T ] with lawμ p (and the curve Z ) are the unique strong solution to (5.1). Trivially, we have u M * ⇀ u , hence we deduce (7.18) once again and, in particular, the asymptotic upper bound This concludes the proof.
As a byproduct, we obtain the following result. (Here we used that the recovery sequence for u * is the constant sequence given by u * , see the proof of Theorem 7.3.)

Mean-field limit for a large number of commercial ships
In this section we study the limit of the problem as N → +∞. For this reason we will stress the dependence of initial data and solutions on N .
8.1. Mean-field limit as N → +∞. In this section, we will use the explicit formula for the velocity correction where we set In what follows, we shall use the symbol * 2 to indicate that the convolution is done with respect to the second variable, i.e., η * 2 ν(x) = R 2 η(x, x − x ′ ) dν(x ′ ).
Step 1. (PDE solved by the empirical measures) In terms of ν c ) .

(8.5)
Let us derive the PDE solved by ν c N (t) in the sense of distributions. 20 Let us fix ξ ∈ C ∞ c ((−∞, T )×R 2 ). By (8.2) and (8.5) we have that Step 2. (Convergence of empirical measures ν c N ) To show compactness of the sequence of curves ν c N ∈ C 0 ([0, T ]; P 1 (R 2 )) we rely on a Arzelà-Ascoli Theorem for metric-valued functions. We split the proof in substeps.
Substep 2.1. (Equiboundedness of supports) By Remark 5.3, we have that max n X N n ∞ ≤ R , where the constant R depending on the initial datum X 0 , the final time T , v N ∞ , r, and K cp . This implies that supp ν c N (t) are contained in the closed ballB R for every t ∈ [0, T ].
We observe that he sequence Z N ∞ is bounded. Indeed, by (8.2), the constant C depending on K gg and the set of admissible controls U (bounded). Taking the norm of Z N and by Grönwall's inequality, we obtain that where the constant R ′ depends on K gg , U , and T . By Remark 5.3, for every r ∈ [0, T ] we have that 8) 20 We use here the duality introduced in Footnote 16.
where the constant C depends on K pg , K pc , Z N ∞ (bounded by R ′ ), max n X N n ∞ (bounded by R ), T , and W . Then the Lipschitz continuity of K cp and (8.8) yield where the constant C additionally depends on E(|Ȳ 0 |). By (8.2) and (8.9), for s ≤ t and n = 1, . . . , N we have that where the constant C depends on the constant obtained in (8.9) and additionally on v N ∞ and r. Using as transport plan between ν c N (s) and ν c N (t) the measure γ = 1 N N n=1 δ (X N n (s),X N n (t)) , we obtain that i.e., the curves ν c N ∈ C 0 ([0, T ]; P 1 (B R )) are equi-Lipschitz with respect to the 1-Wasserstein distance.
Substep 2.3. (Compactness) Since the ballB R is compact, the Wasserstein space P 1 (B R ) is compact too [39,Remark 6.19]. 21 Hence the Arzelà-Ascoli Theorem for continuous functions with values in a metric space guarantees the existence of a curve µ c ∈ C 0 ([0, T ]; P 1 (B R )) and a subsequence N k such that Without loss of generality, we do not relabel this subsequence and denote it simply by N . This does not affect the proof, as in Theorem 8.2 we shall prove uniqueness of solutions for the limit problem.
Step 3. (Convergence of Z N ) We let Z = (Z 1 , . . . , Z L ) be the unique solution to Z ℓ (0) = Z 0 ℓ . As in Substep 4.1 in the proof of Theorem 7.1, we get that Step 4. (Convergence ofȲ N ) Let us consider the SDE We will show thatȲ N converges toȲ . 21 In fact, the curves ν c N ∈ C 0 ([0, T ]; P 1 (R 2 )) take values in a compact set of P 1 (R 2 ) independent of N even under weaker assumptions. This is the case, e.g., when q -moments of ν c N (t) with q > 1 are uniformly bounded, i.e., sup N sup t R 2 |x| q dν c N (t)(x) < +∞ for some q > 1 (this can be proven basing on [39,Theorem 6.9]. A uniform bound on the q -moments follows from the analogous assumption on the distribution of initial data by a Grönwall inequality. Substep 4.1. (Well-posedness of (8.12)) There exists a unique strong solution to (8.12). Indeed, let us consider the drift and the constant dispersion matrix σ = √ 2κ Id 2 , so that Let us observe that b is continuous in t and Lipschitz-continuous in Y (with Lipschitz constant independent of t). Indeed, Z is a continuous curve, while by Kantorovich's duality and t → µ c (t) is a continuous curve in the Wasserstein space P 1 (R 2 ). Moreover, the Moreover, we have that where the last inequality follows from the fact that Z is bounded and µ c (t) has support in the ballB R (0) for every t ∈ [0, T ]. We conclude that where the constant C depends on K pg , K pc , Z ∞ , R . Thus the assumptions of Proposition 2.1 are satisfied. Proposition 2.1 also gives us that where the constant C depends on K pg , K pc , Z ∞ , R ,Ȳ 0 , T , and W . For, we start by noticing that Hence, by (8.2), (8.12), (8.13), and by Kantorovich's duality, we have a.s. for 0 ≤ s ≤ t ≤ T Taking the supremum and the expectation, we deduce that and, by Grönwall's inequality, In particular, the constant C depending also on T . By (8.11) and (8.10), we obtain (8.16).
Step 5. (Limit problem) With (8.10), (8.11), and (8.16) at hand, we are in a position to pass to the limit as N → +∞ in (8.7) and prove that µ c is a distributional solution to   i.e., We divide the proof in substeps. Substep 5.1. (Convergence of initial datum term) By the Lipschitz continuity of x → ξ(0, x) and by Kantorovich's duality, we have that By the assumption on the initial data, we have that for every t. By (8.10) it follows that We start by splitting By the Lipschitz continuity of v , by (8.1), by the Lipschitz continuity of η , and by Kantorovich's duality we have that for every x ∈ R 2 and t ∈ where the constant C depends on v and η . Integrating in time and space and using the fact that |r(x)| ≤ C(1 + |x|), thus it is bounded on the compact support of ξ , we obtain that where the constant C depends on v , η , r, ξ , and T . Moreover, the function x → v η * 2 µ c (t)(x) r(x) · ∇ x ξ(t, x) is Lipschitz-continuous with a Lipschitz constant independent of t and depending on v , η , r, and ξ . For, x → v η * 2 µ c (t)(x) satisfies the latter property, since (8.25) where the constant C depends on v and η . As above, r is bounded on the support of ξ . By the Lipschitz continuity of r and ∇ x ξ , we conclude that the product x → v η * 2 µ c (t)(x) r(x) · ∇ x ξ(t, x) is also Lipschitz-continuous. Thus by Kantorovich's duality we obtain that for every t ∈ [0, T ] where C depends on v , η , r, ξ . Combining (8.22), (8.24), and (8.26), by (8.10) it follows that where the constant C depends on v , η , r, ξ , and T . Substep 5.4. (Convergence of divergence term -II) Let us prove that We start by splitting For the first term in the right-hand side of (8.28), we argue analogously to (8.24) to obtain that where the constant C depends on v , η , K cp , ξ , and T . The only difference consists in the fact that we have K cp * μ p N (t)(x) in place of r(x). For this, we need to observe that In the last inequality, we used the fact that, sinceμ p N (t) is the law ofȲ N (t), where the boundedness follows from the convergence (8.16).
For the second term in the right-hand side of (8.28), we start by observing that K cp is Lipschitz, thus we have for every x ∈ R 2 and t ∈ [0, T ] For the third term in the right-hand side of (8.28), we observe that the function x → v η * 2 µ c (t)(x) K cp * µ p (t)(x) · ∇ x ξ(t, x) is Lipschitz-continuous with a Lipschitz constant independent of t and depending on v , η , K cp , µ p , and ξ . This follows from (8.25), from the fact that ξ is compactly supported, and the inequality obtained as in (8.29). By Kantorovich's duality, Proof. The uniqueness of Z is direct, as the ODE for Z is decoupled from the first two equations. Assume now that µ c i ∈ C 0 ([0, T ]; P 1 (R 2 )), (Ȳ i (t)) t∈[0,T ] for i = 1, 2 are solutions to (8.4) with the same initial data, i.e., where supp(µ c i (0)) = supp(µ c 0 ) ⊂B R . As customary in uniqueness proofs for evolutionary problems, we will temporary neglect the assumption that the initial dataȲ 1 (0) andȲ 2 (0) are a.s. equal and µ c 1 (0) and µ c 2 (0) are equal in order to carry out a Grönwall-type argument to deduce stability with respect to initial data. The objective is to prove the following pair of estimates: These two inequalities provide uniqueness when combined. Indeed, ifȲ 1 (0) =Ȳ 0 =Ȳ 2 (0) a.s. and µ c 1 (0) = µ c 0 = µ c 2 (0), then (8.31) simply reads Substituting into (8.32), we get that which by Grönwall's inequality yields W 1 (µ c 1 (t), µ c 2 (t)) = 0 for all t ∈ [0, T ]. Then (8.31) gives E( Ȳ 1 −Ȳ 2 ∞ ) = 0 .
We divide the proof of (8.31)-(8.32) in several steps. Step |K pc * µ c 1 (r)(Ȳ 1 (r)) − K pc * µ c 1 (r)(Ȳ 2 (r))| dr + s 0 |K pc * µ c 1 (r)(Ȳ 2 (r)) − K pc * µ c 2 (r)(Ȳ 2 (r))| dr . The first integrand in (8.33) is bounded using the Lipschitz continuity of K pg by The second integrand in (8.33) is estimated using the Lipschitz continuity of K pc as follows The third integrand in (8.33) is estimated using the Lipschitz continuity of K pc by Kantorovich's duality Taking the supremum in s and the expectation we deduce that for t ∈ [0, T ] and, in particular, where the constant C also depends on T .
Step 2. (Introducing the flow for the transport equation) Following an idea in [35,36], we prove uniqueness by regarding the solutions of the transport equation from a Lagrangian point of view. Let us consider for every x ∈ supp(µ c i (0)) the flow   [38,Theorem 5.34]. Let us show that the flows Φ i are bounded. We notice that where we used the bound (8.15). By (8.37) and by the estimate |r(X)| ≤ C(1 + |X|) we deduce that for every x ∈B R By Grönwall's inequality and since x ∈B R , we obtain that where the constant C depends on v ∞ , r, K cp , R , and T (in addition to the constant in (8.15)).
In what follows we will show that We start by observing that for every x, x ′ ∈ R 2 and t ∈ [0, T ].
Step 3. (Estimate of |Φ 1 (t, x) − Φ 2 (t, x)|) We estimate the first term in the right-hand side of (8.40) as follows: In the following substeps we estimate the five terms on the right-hand side of (8.41). Substep 3.1. Let us estimate the first term in the right-hand side of (8.41). For x ∈ supp(µ c i (0)) and s, x)) .
These equations allow us to estimate We estimate the first term in the right-hand side of (8.61) using the Lipschitz continuity of H d (y − ·) and Kantorovich's duality by CW 1 (ν c N (s), µ c (s)) .
For the second term in the right-hand side of (8.61), we observe thatȞ d * µ c (t) is Lipschitz continuous, as hence, sinceμ p N (t) is the law ofȲ N (t) and µ p (t) is the law ofȲ (t), where µ c ∈ C 0 ([0, T ]; P 1 (R 2 )), (Ȳ (t)) t∈[0,T ] are obtained by the unique solution to (8.4) and µ p is the law of (Ȳ (t)) t∈[0,T ] . Trivially, we have u N * ⇀ u , hence we deduce (8.60) once again and, in particular, the asymptotic upper bound