Cucker-Smale flocking particles with multiplicative noises: stochastic mean-field limit and phase transition

In this paper, we consider the Cucker-Smale flocking particles which are subject to the same velocity-dependent noise, which exhibits a phase change phenomenon occurs bringing the system from a"non flocking"to a"flocking"state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.


Introduction
In the current work, we are interested in stochastic flocking systems with multiplicative noises in the Stratonovich sense. Let (Ω, F , (F t ) t≥0 , P) be a probability space endowed with a filtration (F t ) t≥0 . Here Ω is the random set, P and F are measure and σ-algebra on the set, respectively. On that probability space, (B t ) t≥0 denotes a real-valued Brownian motion. Let X i t ∈ R d and V i t ∈ R d be position and velocity of i-th particle at time t ≥ 0, respectively, then our main stochastic differential equations read as follows: (1. 1) or, equivalently, in the Itô form, subject to the deterministic initial data (X i 0 , V i 0 ), for i = 1, · · · , N . HereV t is an averaged particle velocity, i.e., V t := 1 N N j=1 V j t and F [µ] represents a velocity alignment force given by where ψ : R + → R + called a communication weight, which is in general non-increasing function. Note that the stochastic particle system (1.1) has locally Lipschitz coefficients, thus the system (1.1) has a strong solutions and pathwise uniqueness holds, see [10,Theorem 3.2]. When there is no noise, i.e., σ = 0, the stochastic particle system (1.1) is reduced to the Cucker-Smale model [8]. We refer to [5,6] for a recent overview of Cucker-Smale and its variants. The system (1.1) is proposed in [1] by taking into account uniform randomness in the communication weight function. More precisely, the system (1.1) can be derived from the original Cucker-Smale model by replacing ψ with ψ + √ 2ση t , where η t is d-dimensional Gaussian white noise. In [1], a flocking estimate showing the relative positions are uniformly bounded in time and relative velocities converge to zero as time goes to infinity almost surely is obtained. Later, in [16], the phase change phenomenon from non flocking to flocking states in (1.1) is observed by considering the convergence of relative velocities in the L 2 -norm. For a flocking estimate of the Cucker-Smale model with N -independent noise of uniform strength, we refer to [12].
Formal passage to the mean-field limit N → ∞ for the particle system (1.1) yields the following stochastic partial differential equation: (1.3) or again equivalently, in the Itô form: (1.4) wherev t := R d ×R d v µ t (dx, dv). The limiting equation (1.3) is indeed stochastic in this case as shown for instance in [7], where a system of interacting particles are subject to the same space-dependent noise is discussed. This is due to the fact that the noise which drives the motion of each particle in (1.2) is the same. In classical McKean-Vlasov particle system, the noise seen by each particle are independent from each other [13,Theorem 1.1], and the limiting equation becomes a deterministic diffusion equation. This result can be classically proved by coupling the N -particle system with independent initial condition to the N independent copies of the nonlinear particle. It is worth emphasizing that the independence of noises in the system is important in that coupling method, see [13] for more details on that.
The first purpose of this paper is to establish the global existence and uniqueness of measure-valued solutions to the stochastic partial differential equation (1.3), and the rigorous analysis of the stochastic mean-field limit of the system (1.1). As pointed out in [7], the equation (1.3) can be understood as a standard transport PDE as the random ω ∈ Ω is fixed. The empirical measure (µ N t ) t≥0 associated to the stochastic particle system (1.1) solves the stochastic partial differential equation (1.3) for any finite N , see Section 2.1 below for details. This enables us to take a strategy based on weak-weak/weak-strong stability estimates used for deterministic transport type equations [3,4,9] for fixed ω ∈ Ω. More precisely, if (µ t ) t≥0 and (ν t ) t≥0 are two solutions to (1.3) for the respective initial data µ 0 and ν 0 ∈ P , we need to establish some (local in time) inequalities of the type: sup where C T is a nonnegative constant depending on the time and other parameters of the problem, or a weaker version sup Here C T is a nonnegative almost surely finite random variable depending on the time and other parameters of the problem. Here W 2 denotes the Wasserstein distance of order 2 defined by where Γ(µ, ν) is the set of all probability measures on R d × R d with first and second marginals µ and ν, respectively, and (X, Y ) are all possible couples of random variables with µ and ν as respective laws. Compared to the classical case of globally Lipschitz and bounded potentials, the force fields in (1.1) are only locally Lipschitz and bounded in velocity and the result by Dobrushin [9] cannot be directly applied. Note that a classical feature of the Cucker-Smale equation with nonnegative communication weight is to keep the speed of particle velocity bounded by the maximal speed at initial state. On the other hand, in the presence of diffusion, that maximum principle does not hold since the Brownian motion can make the velocities as high as wanted with some non-zero probability. In [2], similar Newtonian types of equations with independent standard Brownian motions, which have locally Lipschitz potentials, are considered, and the high speed of particle velocities are controlled by imposing the exponential moments bound. However, in our case for the stochastic transport PDE, we can obtain a P-almost sure propagation of the compact support in velocity if the initial data is compactly supported in velocity. This only gives that the force fields are Lipschitz and bounded P-almost surely, thus we can have a similar inequality as (1.6), but not the type of (1.5) since the Lipschitz constant of force fields is a random variable which does not have any exponential moments, see Proposition 3.1. This stability estimate enables us to approximate a solution µ t to the equation (1.3) by the empirical measure µ N t associated to the particle system (1.1), and in fact, this provides the stochastic mean-field limit. We remark that the mean-field limit of the particle system (1.1) is studied in [11], and a Fokker-Planck type equation is derived as the corresponding mean-field equation. However, that corresponds to the Cucker-Smale model with N -independent Brownian motions, i.e., adding √ 2σ(V t − V i t )dB i t , not the dependent Brownian motion appeared in (1.1).
Our second goal in this paper is to discuss the phase change phenomenon in the limiting stochastic kinetic equation (1.3) showing the transition from non flocking to flocking states as the strength of noises decreases. We notice that flocking behavior of solutions implies the concentration of velocities of particles, i.e., formation of a Dirac delta in velocity, see Remark 1.3. Thus it is natural to consider measure-valued solutions in our notion of solutions for the time-asymptotic behavior of solutions. Since the empirical measures associated to the particle system (1.1) well approximate the measure-valued solutions to the stochastic partial differential equation (1.3), see Proposition 3.1, we can easily extend the result of phase change phenomenon at particle level to the infinite-dimensional one.
Before stating our main results, we first introduce a notion of measure-valued solutions to the kinetic system (1.3). For this, we use a standard notation: φ is an adapted process with a continuous version.
Remark 1.1. The weak formulation in Definition 1.1 can be rewritten as We now state our first result on the global existence and uniqueness of measure-valued solutions to the stochastic partial differential equation (1.3).
be compactly supported in velocity and T > 0. Suppose that the communication weight ψ ∈ C 1 b (R + ). Then there exists at most one measure-valued solution to equation in the sense of Definition 1.1, which is almost surely compactly supported in velocity. Moreover, µ t is determined as the push-forward of the initial density through the stochastic flow map generated by the local Lipschitz field (v, Furthermore, if µ andμ are two such solutions to the equation (1.2) with compactly supported initial data µ 0 andμ 0 in velocity, we have for t ∈ [0, T ] almost surely, where the constant C depends only on ψ, T, σ, sup t∈[0,T ] |B t |, and the support in velocity of µ 0 andμ 0 . Remark 1.2. As mentioned before, the empirical measure t ) associated to the particle system (1.1) is the solution to the stochastic partial differential equation (1.3) in the sense of Definition 1.1, see Section 2.1. Thus it follows from the stability estimate in Theorem 1.1 that Note that we can construct the initial atomic measures µ N 0 approximating the initial data µ 0 such that W 2 (µ 0 , µ N 0 ) → 0 as N → ∞ in the standard way: we define a regular mesh of size 1/N and approximate µ 0 by a sum of Dirac masses µ N 0 located at the center of the regular cells such that the mass at each particle is exactly equals to the mass of µ 0 contained in the associated cell. Then we get W 2 (µ 0 , µ N 0 ) ∼ 1/N → 0 as N → ∞. Our second result on the phase change phenomenon from a "non flocking" to a "flocking" state depending on the strength of noises is presented below. In order to state our theorem, we need to introduce a variance functional of the stochastic particle velocity fluctuation aroundv 0 : For the sake of notational simplicity, we denote by Theorem 1.2. Let µ t be a measure-valued solution for the equation (1.3). Suppose that the communication weight function ψ satisfies 0 < ψ m ≤ ψ(s) ≤ ψ M for s ∈ R + . Then we have

This subsequently implies
where W 1 denotes the Wasserstein distance of order 1 and ρ t is the random spatial probability, i.e., ρ t (x) := R d µ t (dv). Indeed, for any bounded Lipschitz function φ, we find as t → ∞ in probability, due to Theorem 1.2. This, together with the fact that W 1 is equivalent to the bounded Lipschitz distance, concludes the desired result.
Remark 1.4. We can obtain the convergence of the variance functional E t without taking the expectation. More precisely, we find the following almost surely convergence when ψ(s) ≥ ψ m > 0: at least exponentially fast. Find the details of the proof in Proposition 4.1.
The rest of the paper is organized as follows. In Section 2, we show that the empirical measures associated to the particle system (1.1) are solutions to the equation (1.3) in the sense of Definition 1.1. We also present a stochastic Gronwall type inequality which will be used later for the almost surely bound estimate of compact support of solutions in velocity. Using that support bound estimate in velocity together with the weak stability estimate, we provide details on the proof of Theorem 1.1 in Section 3. Finally, we show the phase change phenomenon of the stochastic partial differential equation (1.3), which proves Theorem 1.2, in Section 4.

2.1.
Itô's formula. In this part, we show that the empirical measures associated to the stochastic particle system (1.2) are weak solutions to the stochastic partial differential equation by employing the Itô's formula. For the sake of mathematical simplicity, we work in the corresponding Itô form (1.2). We want to emphasize that this observation implies the limiting system cannot be deterministic since the empirical measures are stochastic for any N . For , if we apply for Itô's formula to the system (1.2), then we obtain Averaging the above equation over i = 1, · · · , N deduces We also easily check that This yields that µ N t associated to (1.2) satisfies the weak formulation in Definition 1.
,··· ,N are solutions to the system (1.1). Thus they are adapted and continuous and this gives that µ N t is a weak solution to the system (1.4) in the sense of Definition 1.1.

2.2.
Stochastic Gronwall type inequality. In this subsection, we provide a stochastic Gronwall type inequality which will be crucially used in this paper.
Lemma 2.1. Let X t be a real value process satisfaying then it holds with probability one.
Proof. Let us denoteȲ t := X 0 e (c1−c 2 2 /2)t e −c2Bt . Using Itô's rule, we find thatȲ t solves We next consider Then again by Itô's rule we get On the other hand, Z t satisfies In this section we establish the global well-posedness of the equation (1.2). Note that we already observed that the empiricial measure associated to the particle system (1.1) is a weak solution to (1.2) in the previous section.
Let us consider two solutions (µ t ) t≥0 and (μ t ) t≥0 to the stochastic partial differential equation (1.4) in the sense of Definition 1.1 with the initial data µ 0 ,μ 0 ∈ P 2 (R d × R d ), respectively, such that Note that we can assume the above without loss of generality due to the conservation of momentum. In particular, this implies Note that the above stochastic characteristics are globally well-defined since the velocity alignment force term is locally bounded and Lipschitz. Applying Itô's rule, it is clear that . We now defineμ t by the push-forward of µ 0 by Z t , i.e.,μ t := Z t #µ 0 . Then this gives that the random probability measure familyμ t solves the following linear stochastic PDE: with the initial dataμ 0 = f 0 . On the other hand, the uniqueness of solutions for that linear stochastic PDE holds, thus we get µ t =μ t = Z t #µ 0 . In the lemma below, we provide a priori kinetic energy and velocity support estimates. Lemma 3.1. Let (µ t ) t≥0 be a solution of (1.3), and (Z t ) t≥0 be the associated stochastic characteristic. Then, it holds with probability one. Moreover, still with probability one, it holds Proof. It follows from (3.1) that Integrating (3.2) with respect to µ 0 (dx, dv), we find This together with Lemma 2.1 gives Coming back to (3.2), we obtain Hence, by Lemma 2.1, we have and this concludes the desired result.
We now provide the stability estimate of solutions in 2-Wasserstein distance in the proposition below.
Proposition 3.1. Let µ t ,μ t be solutions to the equation (1.3) with compactly supported initial data µ 0 ,μ 0 ∈ P 2 (R d × R d ) in velocity, respectively. Then there exists an almost surely finite random variable C T depends only on ψ, T, σ, sup t∈[0,T ] |B t |, and the support in velocity of µ 0 andμ 0 , such that for t ∈ [0, T ], P-almost surely.

This gives
=: J 1 s + J 2 s . Using these newly defined terms, we split I s into two terms: and here I 1 s can be estimated as follows.
where R > 0 is chosen such that supp(µ 0 ), supp(μ 0 ) ⊆ B(0, R) since µ 0 andμ 0 are compactly supported in velocity. We next estimate I 2 s as Combining all the above estimates, we obtain for 0 ≤ t ≤ T almost surely, whereH T := sup 0≤t≤T H t . We now apply Lemma 3.1 with c 1 = 4σ+2ψ M +2 ψ LipHT , c 2 = 2 √ 2σ, and A s = 2 ψ LipHT P s to get where C T > 0 depends only on ψ, σ, sup t∈[0,T ] |B t |, and R. On the other hand, we easily find This together with (3.4) yields and applying Gronwall's inequality gives This completes the proof.
We now define the stochastic empirical measure µ N t associated to a solution to the stochastic particle system (1.1) as with the initial data satisfying where µ 0 ∈ P 2 (R d × R d ) is compactly supported in velocity. On the other hand, P 2 (R d × R d ), W 2 is a complete metric space, thus the sequence µ N 0 is Cauchy. As discussed before, µ N t is a solution to the equation (1.3) and this together with the stability estimate in Proposition 3.1 yields This implies that the (µ N . ) N ∈N converges to some µ . ∈ C [0, T ]; P 2 (R d × R d ) almost surely. Then it remains to prove that (µ t ) t∈[0,T ] solves our main equation (1.2). Note that it follows from Lemma 3.1 that the solution µ t is compactly supported in velocity since the initial data is compactly supported in velocity. For (3.5) We notice that I φ,t (µ N ) = 0 for any N ∈ N almost surely. Thus we find We then claim that , and µ N t and µ t are compactly supported in velocity, we easily obtain that ⋄ Estimate of I N 5 : Note that I N 5 is a linear stochastic integral term. Let us denote by ⋄ Estimate of I N 6 : We now estimate the nonlinear term. We rewrite where the second term on the right hand side goes to 0 as N goes to infinity almost surely since ∇ v φ · F [µ t ] is locally bounded and Lipschitz, and µ N t converges to µ t as N goes to infinity uniformly in time. Indeed, we get as N → ∞, almost surely. For the estimate of the first term, we rewrite it that as In order to treat this last term we use the following lemma.
Then P-almost surely, for any t ∈ [0, T ], the vector field G(µ N t ) is bounded and locally Lipschitz in velocity. Proof. For any µ ∈ P 2 (R d × R d ) we first easily find that for some C > 0. Next for (y, w), (y ′ , w ′ ) ∈ R 2d × R 2d , we obtain where I is easily estimated by.
For the estimate of J, we also easily get Hence it holds This together with Lemma 3.1 completes the proof.
Using Lemma 3.2, since µ N t and µ t are compactly supported in velocity and µ N t converges to µ t almost surely uniformly in time, we have almost surely. Putting all those estimates together deduces that I φ,t (µ) defined in (3.5) is equal to sequences which go to 0 as N → ∞ in probability, thus that is equal to 0 almost surely. This concludes that (µ t ) t∈[0,T ] satisfies the weak formulation in Definition 1.1 (ii). We now show that µ t is weakly continuous. Note that we can always assume that the process ( µ t , φ ) t∈[0,T ] is adapted by changing the notion of the filtration (F t ) t∈[0,T ] . Indeed, that can be obtained as the limit of the sequence of the adapted processes ( µ N t , φ ) t∈[0,T ] in law. Thus it only remains to establish the existence of a continuous version of the process ( µ t , φ ) t∈[0,T ] to complete the proof. Recall that µ t obtained above satisfies In the rest of this section, we estimate the Lebesgue and itô integrals as follows.
⋄ Estimate of the Lebesgue integral: Since φ ∈ C 2 c (R d × R d ) is compactly supported in both position and velocity, we get where C σ,φ is a constant which depends only on σ, φ. This gives On the other hand, it follows from Remark 3.1 that µ s is compactly supported in velocity. Thus we obtain that the alignment force field is almost surely bounded from above by Hence we have that (  In this section, we provide details of the proof of Theorem 1.2 on the flocking and non flocking estimates for the stochastic kinetic equation (1.3). As mentioned in Introduction, we employ a similar strategy used for the particle system (1.1) proposed in [1,16].
Recall the variance functions of stochastic particle velocity fluctuation aroundv t : Note thatv where we used This, together with a straightforward computation, yields Thus the process E t satisfies Taking the expectation to the above gives Note that R 2d ×R 2d |w − v| 2 µ t (dx, dv)µ t (dy, dw) = 2E t , due tov t = 0. Thus we get and this together with (4.2) provides Applying Gronwall's inequality, we have This completes the proof of Theorem 1.2. As mentioned in Remark 1.4, we can also obtain the convergence of the variance functional E t without taking the expectaiton even though it does not provide the phase change phenomenon. Let us go back to equation (4.1). Using the similar strategy as before, we estimate the drift term in (4.1) as Applying Itô's formula gives Taking the time integration to the above inequality gives i.e., On the other hand, by using the fact that for any ǫ > 0, almost surely, there exists t 0 > 0 such that |B t | ≤ ǫt for all t ≥ t 0 , we can further estimate E t ≤ E 0 exp (−2(ψ m − ǫ 0 )t) a.s., for some 0 < ǫ 0 < ψ m and t ≥ t * > 0.
Summarizing the above discussion, we have the following result. Remark 4.1. The stochastic process E t in (4.1) resembles a geometric Brownian motion. Note that the inequality (4.3) can be rewritten as where B ν t denotes the one dimensional Brownian motion with constant drift ν ∈ R, i.e., B ν t = B t + νt. Remark 4.2. When there is no multiplicative noise σ = 0, then the system (1.1) becomes the original Cucker-Smale model. For the Cucker-Smale model, the unconditional flocking estimate can be obtained if the communication weight ψ is not integrable, see [6]. If not, suitable assumptions for the initial configurations are needed for the flocking estimate. In [1], the multiplicative noise is considered for the stochastic particle system (1.1) in the Itô sense, and in that case, we can get the flocking behavior for any nonnegative communication weight ψ, i.e., taking into account the multiplicative noises in the Cucker-Smale model enable us to have the unconditional flocking estimate. This implies that the multiplicative noise in the Itô sense plays a role as a (stochastic) control which enhances the flocking behavior of particles.