Propagation of chaos for the VPFP equation with a polynomial cut-off

We consider a $N$-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like $N^{-\delta}$ with $\delta<1/d$ in the force, we provide a quantitative error estimate between the empirical measure associated to that $N$-particle system and the solutions of the $d$-dimensional Vlasov-Poisson-Fokker-Planck system. We also study the propagation of chaos for the Vlasov-Fokker-Planck equation with less singular interaction forces than the Newtonian one.


Introduction
The starting point is the classical Newton dynamics for point particles interacting through the interaction force F in the presence of noise: i = 1, · · · , N, t > 0, (1.1) where (X t , V t ) := (X 1 t , · · · , X N t , V 1 t , · · · V N t ) ∈ R 2dN are positions and velocities of the particles in R d , and σ > 0 is the noise strength. F (x) = ξ x |x| d where ξ = +1 for plasma problems −1 for astrophysics problems. (1.2) We are interested in the rigorous derivation of the Vlasov-Poisson-Fokker-Planck (VPFP) equation as N → ∞: with the Poisson force F (x) = ξx/|x| d , ξ = ±1, where f = f (x, v, t) is the probability density function at phase space (x, v) at time t. Notice that the VPFP system is the formal mean-field limit equation associated to (1.1). This is a classical kinetic equation whose well-posedness and qualitative properties were studied in different settings: classical solutions [37,3], weak solutions [4,8,9,36,24] and the references therein. We refer to [17,32] for general theory on these problems. The rigorous proof of the mean-field limit in the one dimensional case for (1.3) with the force F given in (1.2) was obtained in [19,23]. In [26], the propagation of chaos for the system (1.3) with bounded forces, i.e., F ∈ L ∞ (R d ) is studied based on relative entropy arguments. The intimately related classical Vlasov-Poisson system has also received lots of attention in the last years. This system corresponds to (1.1) and (1.3) with σ = 0. The propagation of chaos has been shown in [20,21] for the less singular case, i.e., |F (x)| ≤ |x| −α with α < 1. More recently the physically relevant case of the propagation of chaos for the Newtonian potential with polynomial cut-off has been obtained in [27].
The propagation of chaos and its consequence, the rigorous proof of the mean-field limit, are questions of nowadays importance in many other problems; for instance, in kinetic models of collective behavior [2,5,7,10,11,12] with or without noise in velocity, see also [13,30] for more general types of equations. However, in most of these applications the singularity of the kernels is much better behaved. The case of Hölder interaction with Hölder exponent greater than 2/3 was also recently treated in [25]. The main motivation of our present work is to obtain the propagation of chaos in the degenerate diffusion setting, as the noise acts only on velocity, in the classical case of Newtonian interaction with cut-off.
Finally, let us point out that most of the propagation of chaos results for nonlinear nonlocal conservation equations with non Lipschitz interaction, concern first order models. Without diffusion, the mean field limit for the aggregation equation with interaction less singular than the Newtonian is studied in [6]. In one dimension, the case of all power-law singularities were treated in [1] (also with noise) while a corresponding result in higher dimensions was recently obtained in [14,35] for the repulsive case. With diffusion, the case of Hölder interaction is treated in [25] by quantitative means leading to explicit rates of convergence. For the 2D Navier-Stokes equation in vortex formulation, the propagation of chaos is obtained in [16] by compactness, thanks to result of [22]. Similar techniques have been applied to the case of the diffusion dominated 2D Keller-Segel equations in [18,34] and geometrical constraints interactions with reflecting diffusions in [10].
Let us be more precise about the different approximation levels involved in the propagation of chaos. Consider the regularized particle system where a ∨ b := max{a, b}. Note that this system of stochastic differential equations (SDEs) (1.4) has a unique strong solution by the standard theorem for SDEs. It is worth mentioning that the sign of force term is not important for the propagation of chaos provided that we analyze it on a finite time interval. Thus we only focus on the case ξ = 1. Our main purpose is to provide a quantitative error estimates between the solutions to the particle system (1.4) and the VPFP equation (1.3), which extends the work and method in [27] to the case with noise. For this, we need to introduce an intermediate system of independent copies of nonlinear SDEs with cut-off given by , for all i = 1, · · · , N , is the global-in-time weak solution to the regularized VPFP system Due to the regularization, we can easily obtain the global existence of weak solutions to the equation (1.6) under suitable assumptions on the initial data. Solving the system (1.5) with the given ρ N t to get the existence and uniqueness of solutions to the system (1.5) is by now standard.
Before stating our main result in this paper, we need to introduce some notions and notations. We define the empirical measure µ N t associated to a solution to the particle system (1.4) and ν N t the one associated to the nonlinear independent particle (1.5) as respectively. For a function f and p ∈ [1, ∞], f p represents the usual L p -norm for functions in L p (R 2d ), and f p, is the set of L p functions from an interval (0, T ) to a Banach space E. P(R 2d ) and P p (R 2d ) stand for the sets of all probability measures and probability measures with finite moments of order p ∈ [1, ∞). We consider the Wasserstein distance of order p, W p , p ∈ [1, ∞], on P p (R 2d ), see [38] for classical definitions and properties of optimal transport distances. We also denote by C a generic positive constant independent of N .
The main result of this work gives an explicit estimate on the decay of the distance between the empirical measure of the interacting particle system µ N t and the solution of the VPFP equation f t in optimal transport distances in the probability sense.
We next present the existence and uniqueness of weak solutions for the equation (1.3), which gives the all required a priori hypotheses in Theorem 1.1 for the propagation of chaos, see also Lemma 4.1.
Let us briefly explain the strategy of the proof of this theorem.
• First, using the techniques introduced in [27], one can estimate the probability that the error between the empirical measure µ N t associated to the particle system (1.1) and the empirical measure ν N t associated to the nonlinear independent particle system with cut-off (1.4), exceeds the threshold N −γ . In this way, it is shown that this probability decreases faster than any negative power of N (see Lemma 3.2).
• Then using a concentration inequality of [15], one can obtain bounds on the probability that the error between ν N t empirical measure associated to i.i.d. random variables and their law f N t solution at time t to equation (1.6), exceeds the threshold N −γ . The main results in [15] provide an optimal rate of convergence which is of order of some negative power of N . (See Proposition 2.1).
• Finally, we show that the error between the solution to equation (1.6) and the solution to equation (1.3) never exceeds the threshold N −γ , for N large enough. Contrary to [27,Proposition 9.1], this error has to be estimated in W p distance with 1 ≤ p < ∞ due to the presence of noise in velocity (See Proposition 3.1). Our main contributions in comparison to the noiseless case treated in [27] are the following. On the one hand, we provide a convergence result of the solution to equation (1.6) to the solution to equation (1.3), in W p metric for p ∈ [1, ∞), which is crucial since the support of f N t and f t are not compactly supported in our present case. On the other hand, we provide a well-posedness result for equation (1.3). This requires to show that the spatial density ρ t of solution to this equation lies in . For the case without diffusion, i.e., Vlasov-Poisson system, characteristic methods can be used to get a uniform bound on the spatial density, which leads to a uniqueness of solutions, under suitable assumptions on the initial density [28,29,31]. However, the presence of diffusion makes it more complicated. In [33], an uniform-in-time L ∞ -bound of the spatial density is obtained by means of the stochastic characteristic method under the assumptions on compactly supported initial density f 0 in velocity. We also provide a simple proof of the local-in-time L ∞ propagation by employing Feynman-Kac's formula assuming only that the initial data has a polynomial decay in velocity (see Lemma 4.1 together with Theorem 1.2). Notice that obtaining the bound estimate of ρ t in L ∞ (R d ) is equivalent to the one for ρ N t ∞ in N . The rest of this paper is organized as follows. In Section 2, we deal with some preliminary materials introduced in [27]. Section 3 contains the key new estimate on the W p stability between of the solutions of the VPFP equation with and without cut-off. Section 4 shows the well posedness of solution to the VPFP equation with the assumed regularity in Theorem 1.1, i.e., the assumed L ∞ bound on the spatial density. Finally, Section 5 contains generalizations of this result for less singular kernels.

Preliminaries
In this section, we provide estimates for the force fields in the equations (1.3) and (1.6) in L 1 (0, T ; L ∞ (R d )) and the q-th moment estimate for the solutions of that. We also recall several useful estimates on the force fields whose proofs can be found in [27].
Under the assumptions on the spatial densities ρ t and ρ N t in Theorem 1.1, we can easily find These estimates together with straightforward computations yield that for some q ≥ 2 Thus, we obtain and similarly, we also have Let us now recall the interaction force with a cut-off: We drop the subscript δ in F N δ and l N δ for notational simplicity in the rest of paper, i.e., F N δ = F N and l N δ = l N . Lemma 2.1. Let d > 1 be given.
1. There exists a constant C, which depends only on d, such that Proof. See Lemmas 6.1 and 6.3 of [27].
Then we recast here some law of large number like estimates. For κ, δ > 0, we set h : Proof. See the proof of Proposition 7.2 of [27].
We conclude this section by recalling some concentration inequalities in the proposition below whose proof can be found in [15,Theorem 2].
Then, for any p ∈ (0, q/2), ε ∈ (0, q), there exist constants C, c > 0 depending only on d, p, q, ε and the q-order moment bound of ρ such that 3. Propagation of Chaos: Proof of Theorem 1.1 In this section, we provide the details of the proof of Theorem 1.1. We begin with the following Lipschitz estimates on the force fields F and F N .
where C 0 is a positive constant depending only on d.
Proof. The first assertion is straightforward. For the proof of the second one, we consider three cases as follows.
(i) |x|, |y| ≥ N −δ : In this case, we get F N (x) = F (x) and thus it is clear to obtain Note that where C > 0 depends only on d. This yields (ii) |x|, |y| ≤ N −δ : By definition of F N , we find (iii) |x| < N −δ ≤ |y| or |y| < N −δ ≤ |x|: For |x| < N −δ ≤ |y|, let us definex as the intersection between the line segment [x, y] and the ball B(0, N −δ ). Then we get By employing the similar arguments as in the previous cases (i) and (ii), we have This implies that for |x| < N −δ ≤ |y| Similarly, we have Combining the above all cases, we conclude to the desired result.
Proposition 3.1. Let (X, V ) and (Y, W ) be two random variables of law f 1 and f 2 respectively, such that their first marginal are ρ 1 and ρ 2 , respectively. Let (X, Y ) be an independent copy of (X, Y ). Suppose ρ 1 , ρ 2 ∈ L ∞ (R d ) and N ≥ e. Then we have for p ≥ 1, where ln − (x) := ln(x) ∧ 0 and the constant C > 0 depends only on d.
Proof. Let us denote by α d the surface area of unit ball in R d .
Estimate for the non cut-off force field.-First, we notice that This yields that for any r > 0 Estimate of I 1 : Note that the event {|X − X| ∧ |Y − Y | ≤ r} can be partitioned as Taking into account this, we split I 1 into three terms: For the estimate of I 1 1 , by separating in the notations the expectation w.r.t. the independent copies (X,Ȳ ), from the expectation w.r.t. the random variables (X, V ), (Y, W ), we get For I 2 1 , we obtain Similarly, we estimate I 3 1 as I 3 1 ≤ 2α d ρ 1 ∞ rE G p−1 . Combining the above estimates, we have where C > 0 only depends on d.
Estimate of I 2 : We decompose I 2 as First we easily obtain We then consider two cases: r > 1 and 0 < r ≤ 1. For r ≤ 1, we get For the case r > 1, it is clear to obtain This yields where C > 0 only depends on d. For the term I 2 2 , we use Hölder inequality to find Similarly as before, we take the expectations on (X, Y ) and (X, Y ) for the first and second expectations in the above, respectively, to find where C > 0 only depends on d. Thus, by putting all those estimates together and using Hölder inequality, we have for any r > 0 Finally, we choose r = E [G p ] 1/p to obtain the desired result. Estimate for the cut-off force field.-Note that since F N is continuous we have As in the proof above, we first easily get where we used Lemma 2.1 and √ ln N |X − Y | ≤ G N . For the term J 2 , we again use the similar argument as before to find Taking the expectations on (X, Y ) in the first expectation and on (X, Y ) in the second one leads to the desired result.
We next estimate the error between solutions to the nonlinear SDEs and the one with cut-off given by

1)
• Nonlinear SDEs with cut-off: Here ρ N t = R d f N t dv and f N t is the global-in-time weak solution of the equation (1.6). As mentioned in Introduction, for fixed N > 0, the global existence and uniqueness of solutions to (3.2) is ensured due to classical SDEs theory. At the moment, we assume the existence of solutions to the SDEs (3.1) and its associated PDE (1.3) up to a given time T > 0. We will give the details of that in Section 4. Proposition 3.2. For a given T > 0, let (Y t , W t ) and (Y N t , W N t ) be the solutions to the equations (3.1) and (3.2) for the same initial condition on the time interval [0, T ], respectively. Suppose that ρ t , ρ N t ∈ L 1 (0, T ; L ∞ (R d )) and ρ N t L 1 (0,T ;L ∞ ) ≤ C with C > 0 independent of N . Then, for N ≥ e and p ≥ 1, we have where C is a positive constant independent of p and N .
Proof. For p ≥ 1, we set Taking the expectation on both sides of the above inequality and using Fubini's Theorem, we obtain where we can directly use the cut-off force field estimate in Lemma 3.1 to estimate I 2 as where C > 0 only depends on d. For the estimate of I 3 , we easily find

This yields
where we used Young's inequality for the last inequality. We then combine the above estimates to have where C > 0 only depends on d. We now apply Lemma A.1.1 with due to the uniform bound assumption on both ρ t and ρ N t in L 1 (0, T ; L ∞ (R d )). This completes the proof.
Remark 3.1. Note that for any p > 0 Let us define the functional J N : R 4dN → R + by Then by using the same argument as in [27,Theorem 4.2], which is based on Gronwall lemma, Lemma 2.2 and Markov's inequality, we obtain the following estimate on J N .
be a solution to (1.4) and let (Y N t , W N t ) t≥0 be solutions to (1.5) with the same independent identically distributed initial conditions. Assume that Then, for any δ ∈ (0, 1 d ) and β > 0, there exists C β > 0 such that This implies Note that under the event {J N t < 1} we get W ∞ (µ N t , ν N t ) < N −δ , thus by using Lemma 3.2, we obtain For the estimate of last term in (3.3), we use Proposition 2.1 with x = N −pγ to have Combining the above estimates concludes the desired result.

Well-posedness of nonlinear SDEs and PDE: Proof of Theorem 1.2
In this section, we study the well-posedness of nonlinear SDEs (1.7) which is associated to the VPFP equation (1.3). For this, we use the nonlinear SDEs with cut-off given by where ρ N t is the spatial density of solution to (1.6). We first show the uniform-in-N estimate of spatial density ρ N t in L 1 (0, T ; L ∞ (R d )). Lemma 4.1. Let T > 0. Assume that the initial data f 0 satisfies f 0 ∈ (L 1 ∩ L ∞ )(R 2d ). Then there exists a unique weak solution f N t to the system (1.6) with the initial data f 0 , such that f N ∈ L ∞ (0, T ; (L 1 ∩ L ∞ )(R 2d )). Furthermore, if we assume that for some C > 0 where ρ N t denotes the spatial density of the law of solution at time t to equation (4.1).
Proof. Since the existence and uniqueness of solutions f N t to the equation (1.6) is classical due to the regularity of the force fields, we only focus on the uniform-in-N estimate of the spatial density ρ N t in the rest of the proof. We divide the proof into two steps. • Step A (Feynman-Kac's representation formula) Let (χ ε ) ε>0 be a family of mollifying kernels. First, we notice that f N t = L(Y N t , W N t ), the law of solution to (4.1), is a solution in the sense of distributions to with the initial data f 0 = L(Y N 0 , W N 0 ). Denote by f N,ε t the solution to the same equation with initial condition f 0 * χ ε . Since the coefficients of the above equation are Lipschitz and locally bounded, classical existence theory guarantees the global existence and uniqueness of strong solutions. We now fix t ≥ 0 and consider the following "backward" stochastic integral equations: It is classical that there exists a unique strong solutions to the above equations due to the strong regularity of the force fields. We next set and apply Ito's rule to θ to find Taking the expectation to the above equation together with Thus, finally, we choose s = t to have Step B (Uniform-in-N estimate) It follows from the previous bound that Note that for all v, w ∈ R d and γ ≥ 1 Using those facts, we get and further we find for γ > d Due to F N * ρ N,ε (x) ∞ ≤ C ρ N,ε 1,∞ , we find for any fixed time T > 0 where C > 0 is independent of N . Finally, we use Lemma A.1.3 to have where C > 0 is independent of N . The result follows from the fact that ρ N,ε t converges at least weakly star to ρ N t in L ∞ (R d ) as ε goes to 0.
We are now ready to give the details of the proof of Theorem 1.2.
Proof of Theorem 1.2. We split the proof into three steps. • Step A (Cauchy estimates) Let (Y N t , W N t ) be the strong solution to the system (4.1) on the time interval [0, T ]. Then by using a similar argument as in Proposition 3.2, we can show that for where C is a positive constant independent of p, N and N .
• Step B (Existence) It follows from the previous step that the sequence (Y N t , W N t ) N ∈N of solution to (4.1) is a Cauchy sequence. Thus there exists a limit process (Y t , W t ) t∈[0,T ] such that (Y N t , W N t ) goes in law to (Y t , W t ) as N goes to infinity. Moreover, denoting by (f N t ) N ∈N the sequence of the law of solution to (4.1), we find where C is a positive constant independent of p and N . This deduces that (f N t ) N ∈N converges weakly to some f t ∈ C([0, T ]; P p (R d )) which is the law of (Y t , W t ) t∈[0,T ] . It now remains to prove that this process is indeed a solution to (1.7). In order to check this, it is sufficient to prove that (F N * ρ N t (Y N t )) t∈[0,T ] converges P almost surely (up to a subsequence) to (F * ρ t (Y t )) t∈[0,T ] . It follows from Proposition 3.1, Lemma 4.1, and ( Set Q(t) := E [∆ p t ], then we get

Vlasov-Fokker-Planck equation with less singular interactions than Newtonian
The previous strategy can directly be applied for the system (1.3) with milder singular interaction forces(see also [27,Section 10] for the case without noise). To be more precise, let us consider the following nonlinear Vlasov-Fokker-Planck equation with singular interactions: where F α satisfies with F α (0) = 0 by definition. Note that α = d − 1 corresponds to the Newtonian case (1.2). Concerning the particle approximations, in a similar fashion as before, we consider the following stochastic particle system with cut-off given by where the cut-off interaction potential F N δ,α is given by F N δ,α (x) = F α (x) for |x| ≥ N −δ and satisfies F N δ,α (x) ≤ N αδ and ∇F N δ,α (x) ≤ N (α+1)δ for |x| < N −δ . Then defining the associated empirical measure we can state the following result.
Let f t and f N t be the solutions to the nonlinear Vlasov-Fokker-Planck equation (5.1) and its corresponding regularization (1.6) with F N δ,α instead of F N δ respectively, up to time T > 0. Assume that 0 ≤ α < d/ − 1 and that f, f N ∈ L ∞ (0, T ; (L 1 ∩ L )(R 2d )) ∩ C([0, T ]; P q (R 2d )) with the same initial data f 0 ∈ (L 1 ∩ L ∩ P q )(R 2d ) for some q ≥ 2 and 1/ + 1/ = 1. Furthermore, assume that their respective spatial densities ρ t and ρ N t satisfy where C 0 only depending on the initial data and T . Then, for any p ∈ [1, 2q), δ satisfying either and δ < d , and ε ∈ 0, q − p 1−pδ , the estimate for N large enough holds for some constant C > 0 depending only on d, T, p, q, ε, α, , f 0 , and C 0 .
Remark 5.1. Using almost same argument as in the proof of Theorem 1.2, we can obtain the wellposedness of the nonlinear SDEs (3.1) and the corresponding PDE (5.1) with F α instead of F under the assumption on the initial data: for α ∈ [0, d − 1). Similarly as before, for notational simplicity, we omit the subscript δ in F N δ,α and l N δ,α in the rest of this section, i.e., F N α = F N δ,α and l N α = l N δ,α . In the lemma below, we provide the weak-strong gradient estimate and uniform bound estimate of the gradient of force field in the cut-off parameter N , which can be obtained in the same manner as Lemma 2.1.
Then slightly modifying the proof of Lemma 2.2, we have the following lemma.
Due to the milder singularity in the interactions than the Newtonian, we can bound the gradient of the force term uniformly in N in Lemma 5.1. This also enables us not to introduce the different weights in position and velocity for the error estimate between solutions of the corresponding nonlinear SDEs and the one with cut-off. More precisely, we have the following proposition which corresponds to Proposition 3.2.
Proposition 5.1. For a given T > 0, let (Y t , W t ) and (Y N t , W N t ) be the solutions to the equations (3.2) with F N α and (3.1) with F α appeared in (5.1) on the time interval [0, T ], respectively. Suppose that ρ t , ρ N t ∈ L 1 (0, T ; L (R d )) and ρ N t L 1 (0,T ;L ) ≤ C with C > 0 independent of N . Then, for p ≥ 1 and d > (α + 1) , we have where C is a positive constant independent of p, α, , and N .
In the lemma below, we provide the estimate onJ N whose proof can be obtained by using the similar argument as in Lemma 3.2.
Lemma 5.3. Let (X N t , V N t ) t≥0 be a solution to (5.2) and let (Y N t , W N t ) t≥0 be N copies solutions to (1.5) with F N α instead of F N and the same independent identically distributed initial conditions. DefineJ N t byJ N t :=J(X N t , V N t , Y N t , W N t ). Suppose d > (α + 1) and ρ N L 1 (0,T ;L ) ≤ C. Then for any β > 0 there is a constant C β > 0 such that it holds P sup Proof of Theorem 5.1. The strategy of the proof is the same as the one of Theorem 1.1. We first estimate W p (µ N t , f t ) as W p (µ N  2. We first claim that f (t) < 1 for t ∈ [0, T ].
We now let t → T * in the above inequality to find 1 = lim t→T * f (t) ≤ exp 1 − (1 − ln f 0 )e −ct < 1, due to the assumption on the initial data f 0 . This is a contradiction that T * < T and implies T * = T . Hence we have f (t)
This yields