On Stochastic Cucker-Smale flocking dynamics

We present a stochastic version of the Cucker-Smale flocking dynamics based on a markovian $N$-particle system of pair interactions with unbounded and, in general, non-Lipschitz continuous interaction potential. We establish the infinite particle limit $N \to \infty$ and identify the limit as a solution with a nonlinear martingale problem describing the law of a weak solution to a Vlasov-McKean stochastic equation with jumps. Moreover, we estimate the total variation and Wasserstein distance for the time-marginals from which uniqueness in the class of solutions having some finite exponential moments is deduced. Based on the uniqueness for the time-marginals we prove uniqueness in law for the Vlasov-McKean equation, i.e. we establish propagation of chaos.


Cucker-Smale flocking dynamics
Cucker and Smale postulated in [CS07b,CS07a] a model for the flocking of birds where convergence to a certain consensus (here same direction and velocity in the motion of birds) was shown to depend on the spatial decay of the communication rate between the birds. In abstract mathematical notation, the Cucker-Smale model describes dynamics of N particles pr k , v k q P R 2d , where r k stands for the position and v k for the velocity of the particle with number k " 1, . . . , N . The time evolution is described by the system of ordinary differential equations ψpr k´rj qpv j´vk q . (1.1) Here ψ ≥ 0 is a symmetric function and describes the communication rate between the particles.
The particular form of (1.1) implies that the mean velocity is conserved, i.e. v c :" Based on Lyapunov functional techniques corresponding to certain dissipative differential inequalities, the time-asymptotic flocking property lim tÑ8 N ÿ k"1 |v k ptq´v c | 2 " 0 and sup t≥0 N ÿ k"1 |r k ptq´r c ptq| 2 ă 8 was studied in [HL09b], where r c ptq :" 1 N ř N k"1 r k ptq " r c p0q`tv c denotes the center of mass. In many cases one seeks to study properties of the particle dynamics in terms of their associated mean-field equations. For the classical Cucker-Smale dynamics the corresponding mean-field equation was derived from the BBGKY-hierarchy when taking the infinite particle limit N Ñ 8 in [HT08]. It was shown that the resulting particle density µ t pdr, dvq solves the kinetic equation ( Existence and uniqueness for measure solutions to (1.2) was established in the class of states where µ t has compact support for each t (see [HL09b]). For different aspects of this model we refer to [She08,HL09a], while other related models are studied in [AH10], [PRT15], [HJN`17], [CHZ18].

Stochastic Cucker-Smale flocking dynamics
In this work we propose a stochastic version of the Cucker-Smale model where, roughly speaking, Bpµq in (1.3) is replaced by a pure jump operator of mean-field type in the velocity component. Let N ≥ 2 be the number of interacting particles x k :" pr k , v k q P R 2d , k " 1, . . . , N . Each particle, say pr k , v k q, may interact with another particle, say pr j , v j q, and the interaction results in a transition of velocities v k Þ ÝÑ v k`p v j´vk`u q " v j`u , (1.4) where u P R d is distributed according to a symmetric probability distribution apuqdu. The rate of this event is supposed to be proportional to ψpr k´rj qσpv k´vj q, where ψ, σ ≥ 0 are symmetric functions on R d . More precisely, consider a Markov process with phase space R 2dN given, for F P C 1 c pR 2dN q, by the Markov generator pLF qpx 1 , . . . , x N q " N ÿ k"1 v k¨p ∇ r k F qpx 1 , . . . , x N q 1 N N ÿ k,j"1 ψpr k´rj qσpv k´vj q ż R d pF px 1 , . . . , pr k , v j`u q, . . . , x N q´F px 1 , . . . , x N qapuqdu.
The following are our minimal conditions assumed throughout this work: (A) ψ ≥ 0 is continuous, bounded and symmetric.
(C) a ≥ 0 is a symmetric probability density on R d .
For most of the results we also assume that a has some finite moments, i.e. holds for some p ≥ 0. The precise value of p will be specified in the corresponding statements. In Section 3 we will prove that the corresponding martingale problem for the generator L with domain C 1 c pR 2dN q is well-posed (see Theorem 2.4). Moreover, we provide estimates on the moments of this process with constants independent of N .

The mean-field stochastic Cucker-Smale dynamics
For each N ≥ 2, let pR N k , V N k q k"1,...,N be the Markov process with phase space R 2dN and generator L. In this work we study the infinite particle limit N Ñ 8 for the sequence of empirical measures Denote by PpR 2d q the space of probability measures over R 2d . We prove in Section 3 that each limit of µ pN q solves a the nonlinear martingale problem with Markov generator pApνqgqpr, vq " v¨p∇ r gqpr, vq`ż R 3d pgpr, w`uq´gpr, vqq ψpr´qqσpv´wqνpdq, dwqapuqdu, (1.6) in the following sense: Definition 1.1. Let µ 0 P PpR 2d q. A solution to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q is a probability measure µ on the Skorokhod space DpR`; R 2d q such that the following conditions are satisfied (i) µpxp0q P¨q " µ 0 .
(ii) It holds that sup sPr0,ts E µ p|vpsq| γ q ă 8, @t ą 0. (1.7) where E µ denotes the expectation with respect to µ and pr, vq is the canonical coordinate process on the Skorokhod space.
is a martingale with respect to µ, where µ s denotes the time-marginal of µ.
It is possible to write the law µ also as a weak solution to a certain Vlasov-McKean stochastic equation (below always called mean-field SDE) specified in the following definition.
Definition 1.2. A process pR, V q is a weak solution to the mean-field SDE, if there exists (i) A stochastic basis pΩ, F, pF t q t≥0 , Pq with the usual conditions.
The next lemma shows that each solution µ to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q can be represented as a weak solution to the mean-field SDE (1.9).
Lemma 1.3. The following assertions hold.
(a) Let pR, V q be a weak solution to the mean-field SDE (1.9) satisfying sup sPr0,ts Ep|V psq| γ q ă 8, @t ą 0. (1.10) Then the law of pR, V q on the Skorokhod space DpR`; R 2d q solves the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q. (b) Let µ be a solution to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q. Then there exists a weak solution pR, V q to the mean-field SDE (1.9) such that pR, V q has law µ.
A proof of this Lemma is given in the appendix. Set xuy :" p1`|u| 2 q 1 2 , u P R d . This function satisfies the elementary inequalities xu`wy ≤ ? 2 mintxuy`xwy, xuyxwyu. (1.11) The main result of Section 3 is summarized in the following existence result for the mean-field model.
Then there exists a weak solution pR, V q to the mean-field SDE (1.9). Moreover, there exists a constant C " Cpψ, σ, a, pq ą 0 such that and, there exists another constant C 1 " C 1 pψ, σ, aq ą 0 such that, for γ P r0, 2s and t ≥ 0, In [FRS18a] we have recently studied the mean-field limit for the Enskog process describing the time-evolution of a gas in the vacuum. The operator Apνq defined in (1.6) is less singular then its analogue considered in [FRS18a]. However, we have not been able to prove that Apνq maps compactly supported functions onto bounded functions (unless γ " 0). Hence in order to indentify the limits of the empirical measures with solutions to a nonlinear martingale problem additional approximation arguments are required.
The following remark shows that the stochastic Cucker-Smale model still satisfies conservation of momentum.
Remark 1.5. Using the particular form of the operator Apνq in (1.6) and the symmetry of σ, a, it is not difficult to see that pR, V q satisfies sup tPr0,T s Ep|Rptq|q ă 8 for all T ą 0, and A similar statement holds also for the particle dynamics.
Sections 4 and 5 are devoted to the study of uniqueness for the mean-field model (uniqueness for the nonlinear martingale problem and uniqueness in law for the mean-field SDE (1.9). Below we formulate only a particular case where σ is bounded from which we are able to deduce propagation of chaos, i.e. convergence of the empirical distributions µ pN q of the particle dynamics.
Then there exists a unique weak solution pR, V q to the mean-field SDE (1.9). Let µ be the law of pR, V q. Then in law on the space of probability measures over the Skorokhod space DpR`; R 2d q.
Convergence (1.14) is a consequence of the uniqueness in law for the mean-field SDE (1.9) and the considerations of Section 3. This convergence is also equivalent to the propagation of chaos (see [Szn91]).
Remark 1.7. The moment condition ş R 2d |v| 4 µ 0 pdr, dvq ă 8 is to strong and can be replaced by ş R 2d |v| 2 µ 0 pdr, dvq ă 8. Indeed, if γ " 0, then we may easily show that the particle dynamics studied in Section 2 preserves second moments with a constant independent of N . Moreover, the proofs given in Sections 3 and 4 remain valid in this case, which implies the assertion of Theorem 1.6.
In the particular case where σ is bounded, we may also prove that the unique solution propagates exponential moments.
Then there exists a unique weak solution pR, V q to the mean-field SDE (1.9), and this solution satisfies for some constant C ą 0.

Structure of the work
This work is organized as follows. In Section 2 we first prove some Lyapunov estimates for the particle dynamics. Then we construct the corresponding Markov process for the particle dynamics and give provide useful moment estimates. Section 3 is devoted to the infinite particle limit N Ñ 8 where Theorem 1.4 is proved. Uniqueness for the case γ " 0 is studied in Section 4 from which we deduce Theorem 1.6 Some further uniqueness results applicable also for the case γ P p0, 2s are studied in Section 5, i.e. we prove estimates on the total variation and Wasserstein distance for the time-marginals of solutions to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q. The proof of Lemma 1.3, some nonlinear generalization of the Gronwall lemma and a localization argument for martingale problems with unbounded generators are discussed in the appendix.
2 The particle dynamics 2.1 Lyapunov estimates for the particle dynamics Let N ≥ 2 be fixed. The following is one of our main estimates for the moments of the particle system.
Lemma 2.1. Suppose that (1.5) holds for some p ≥ 2. Then there exists a constant C " pψ, σq ą 0 such that Since the proof is elementary and not very interesting we postpone it to the appendix. Another useful moment estimate is given in the next lemma.
Lemma 2.2. Suppose that (1.5) holds for p ≥ 1 2 . Then there exists a constant C " Cpψ, σq ą 0 such that Proof. By the mean-value theorem and (1.11) we finďˇx v j`u y 2p´x v k y 2pˇ≤ C2 2p xuy 2p`x v j y 2p`x v k y 2p˘.
Hence we obtain where we have used the Young inequality Finally we give an estimate on the exponential moments.

Well-posedness of the martingale problem
Fix N ≥ 1. It is useful to give a pathwise description of the Markov process associated to L in terms of stochastic differential equations. Namely take a Poisson random measure N on R`ˆt1, . . . , N u 2ˆRdˆR`w ith compensator p N pds, dl, dl 1 , du, dzq " ds b¨1 N N ÿ j,k"1 δ j pdlq b δ k pdl 1 q‚b papuqduq b dz (2.2) defined on a stochastic basis pΩ, F, pF t q t≥0 , Pq with the usual conditions. The law of the Markov process associated to L should then provide a weak solution to the system of stochastic equations where e l " p0, . . . , 0, 1, 0, . . . , 0q P R dN with the 1 placed on the l-th place and GpR, V, u, l, l 1 , zq " e l pu`V l 1´V l q½ r0,ψpR l psq´R l 1 psqqσpV l ps´q´V l 1 ps´qqs pzq. (2.4) Let PpR 2dN q be the space of all probability measures on R 2dN . If σ is bounded, then weak existence and uniqueness in law for (2.3) can be shown by classical localization arguments (see e.g. [EK86]). Below we prove a more general statement including all γ P r0, 2s. Since in such a case LF is not bounded, even if F P C 1 c pR 2dN q, the desired result does not immediately follows from the classical theory of martingale problems [EK86]. Some additional approximation arguments, combined with moment estimates, are required, i.e. we apply Theorem 6.4 from the appendix.
Theorem 2.4. Suppose that (1.5) holds for p :" 2. Then for each ρ P PpR 2dN q with the martingale problem pL, C 1 c pR 2dN q, ρq has a unique solution and this solution can be obtained from a weak solution to (2.3).
Proof. Let g P C 8 pR`q be such that ½ r0,1s ≤ g ≤ ½ r0,2s and set Let L m be the Markov operator given by L with σpv k´vj q replaced by g m pvqσpv k´vj q. Then for each F P C 1 b pR 2dN q we can find a constant C " CpF, ψ, σq ą 0 (independent of m) such that Step 1. Let pΩ, F, F t , Pq be a stochastic basis and let pRp0q, V p0qq P R 2dN be a random variable with some given law µ P PpR 2dN q. Let N m be a Poisson random measure on Ω with compensator . . , N u 2ˆRdˆr 0, c m s (for some constant c m ą 0 large enough). Then p N m pp0, tsˆt1, . . . , N uˆR dˆr 0, c m sq ă 8, @t ą 0 and hence the system of stochastic equations l 1 ps´qqs pzq can be uniquely solved from jump to jump. From [Kur11] we conclude that the martingale problem pL m , C 1 c pR 2dN q, µq has, for each µ P PpR 2dN q, a unique solution whose law can be obtained from (2.7).
Step 2. Suppose that pRp0q, V p0qq has law ρ satisfying (2.5). In order to apply Theorem 6.4 it suffices to show that EpxV m psqy 4 q ă 8, @t ą 0. (2.8) Using the Itô formula and Lemma 2.1 we deduce for some constant C " Cpψ, σq ą 0 (independent of m) and likewise we deduce from Lemma 2.2 This proves (2.8). Hence we may apply Theorem 6.4 to conclude that the martingale problem for pL, C 1 c pR 2dN q, ρq has a unique solution P ρ which satisfies where E ρ denotes the integration w.r.t. P ρ and prptq, vptqq the coordinate process in the Skorokhod space DpR`; R 2dN q.
Step 3. By construction of P ρ we see that, for any F P C 1 c pR 2dN q, is a martingale with respect to P ρ . In view of (2.6) we conclude that (2.9) is a local martingale for any F P C 1 b pR 2dN q. Existence of a weak solution pR, V q to (2.3) having the prescribed law P ρ can be now obtained from [HK90, Theorem A.1].
Remark 2.5. Suppose that ψ and σ are locally Lipschitz continuous. Then similar arguments to [Gra92] can be used to prove strong existence and pathwise uniqueness for (2.3).
We call ρ P PpR 2dN q symmetric, if for any permutation τ of t1, . . . , N u and any bounded measurable function F : The following corollary shows that the particles trajectories are indistinguishable.
Denote by X N k :" pR N k , V N k q, k " 1, . . . , N , the unique weak solution to (2.3). Then X N 1 , . . . , X N N are exchangeable as elements in DpR`; R 2d q, i.e. for any permutation τ of t1, . . . , N u and any bounded measurable function F : (2.10) In particular, pR N k , V N k q, k " 1, . . . , N , are identically distributed as elements in DpR`; R 2d q. Proof. Since L maps symmetric functions onto symmetric functions, the assertion follows from uniqueness of the martingale problem pL, C 1 c pR 2dN q, ρq.

Moments of the particle dynamics
Fix N ≥ 1. Below we prove some moment estimates (uniform in N ) for the unique solution to (2.3).
Corollary 2.7. Suppose that (1.5) holds for some p ≥ 2 and let ρ P PpR 2dN q be symmetric with where C p " Cλ 2p 2 5p , and, for γ " 2, Moreover, there exists another constant C 1 " C 1 pψ, σq ą 0 such that Proof. It follows from Lemma 2.1 and the Itô formula that where we have used the Jensen inequality. Next observe that, by 1`|v| 2p ≤ xvy 2p and previous estimate, ds.
For γ " 2 we apply the Gronwall lemma, for γ P r0, 2q we may apply a nonlinear version of the Gronwall lemma stated in the appendix. Finally assertion (2.12) follows by the Itô formula, similar arguments to Lemma 2.2 and Corollary 2.6.
To be more rigorous one has to consider the above estimates first for the variables V N,m ptq :" V N pt^τ m q where τ m is a stopping time choosen in such a way that V N pt^τ m q is bounded. Obtaining the desired estimates for V N,m ptq (with constants independent of m), one may then pass to the limit m Ñ 8. Since such type of arguments are rather standard, we leave the details for the reader.
Using similar arguments and Lemma 2.2 we can show propagation of exponential moments.

The infinite particle limit N Ñ 8
In this section we perform the limit N Ñ 8 and identify the corresponding limiting process, i.e. we prove Theorem 1.4. Corollary 1.8 can be deduced by the same arguments but now using the moment estimates from Corollary 2.8. For each N ≥ 2, let ρ pN q P PpR 2dN q be given by and denote by pR N k , V N k q k"1,...,N the unique weak solution to (2.3) defined on a stochastic basis pΩ N , F N , pF N t q t≥0 , P N q with the usual conditions. Denote by PpDpR`; R 2d qq the space of probability measures over the Skorokhod space DpR`; R 2d q and, similarly let PpPpDpR`; R 2d qqq be the space of probability measures over PpDpR`; R 2d qq. Define a sequence of empirical measures i.e. random variables with values in PpDpR`; R 2d qq and denote by π pN q P PpPpDpR`; R 2d qqq the law of µ pN q . The proof consists of the following two steps Step 1. Prove that π pN q is relatively compact and show that each limit is supported on processes having the desired moment bounds.
Step 2. Prove that each limit π p8q of a subsequence of π pN q is supported on solutions to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q.

Compactness and moment estimates
Let us show that pπ pN q q N ≥2 is relatively compact.
Proof. In view of [Szn91, Proposition 2.2], see also Corollary 2.6, it suffices to show that where the right-hand side is finite due to the moment estimates of previous section. We seek to apply the Aldous criterion (see e.g. [JS03]). For each N ≥ 2 let S N , T N be pF N t q t≥0 stopping times such that for M P N and δ P p0, 1s we have S N ≤ T N ≤ S N`δ and S N , T N ≤ M . Then and similarly by (2.3) Since 2p ≥ maxt4, 1`2γu, the moment estimates of previous section imply that the right-hand sides are finite. This proves the assertion.
The next lemma provides moment estimates for the limits of the empirical measure.
Lemma 3.2. There exists a constant C " Cpψ, σq ą 0 such that for all t ≥ 0 we have, for γ P r0, 2q, with C p " Cλ 2p 2 5p and, for γ " 2, Proof. By approximation and the Lemma of Fatou we get ż The assertion now follows from Corollary 2.7.
From this we readily deduce, after we have completed Step 2 and Step 3, the desired moment estimates (1.12). Estimate (1.13) follows from the Itô formula and a direct computation.

Identifying the limit
The following shows that each limit point π p8q of a subsequence of pπ pN q q N ≥2 is supported on solutions to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q.
The rest of this section is devoted to the proof of this proposition. It is not difficult to see that the complement of is at most countable and the coordinate function pr, vq Þ ÝÑ prptq, vptqq is µ-a.s. continuous, for any t P D µ and any µ P PpDpR`; R 2d qq. Moreover, we can show that also the complement of is at most countable. Let 0 ≤ t 1 , . . . , t m ≤ s ≤ t with t 1 , . . . , t m , s, t P Dpπ p8q q, m P N, g 1 , . . . , g m P C b pR 2d q and g P C 1 c pR 2d q. For pr, vq P DpR`; R 2d q and µ P PpDpR`; R 2d qq set It is clear that µ is a solution to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q, provided µpxp0q P¨q " µ 0 , (1.7) holds and F pµq " 0. Since, by Lemma 3.2, π p8q -a.a. µ satisfy (1.7) and µpxp0q P¨q " µ 0 , it suffices to show that (a) lim N Ñ8 ş PpDpR`;R 2d qq |F pµq| 2 dπ pN q pµq " 0, where for simplicity of notation π pN q denotes the subsequence converging weakly to π p8q . Let us first prove (a).
Proof. Let r N pds, dl, dl 1 , du, dzq be the compensated Poisson random measure on R`ˆt1, . . . , N u 2R dˆR`w ith compensator given by (2.2), Gpr, v, u, l, l 1 , zq " e l pu`v l 1´v l q½ r0,ψpr l´rl1 qσpv l´vl1 qs pzq, (3.4) where pr, vq P R 2dN , z P R`, u P R d and pl, l 1 q P t1, . . . , N u 2 is defined as in (2.4) and set where E :" t1, . . . , N u 2ˆRdˆR`a nd G k " G k pR N pτ q, V N pτ´q, u, l, l 1 , zq. Then and from the Itô formula one immediately obtains This shows that For the Doob-Meyer process of M N,k s,t we obtain xM N,k s,t y " which implies, in view of the moment estimates of previous section, E N pxM N,k s,t yq ≤ C for all k " 1, . . . , N where the constant C " Cpψ, σ, a, gq is independent of N . Using the particular form of G defined in (3.4), we obtain for the covariation process xM N,k s,t , M N,j s,t y " 0 for all k ‰ j.
Hence we conclude from the properties of the processes xM N,k s,t y and xM N,k s,t , M N,j s,t y ż PpDpR`;R 2d qq |F pνq| 2 dπ pN q pνq which proves the assertion.
Next we prove that assertion (b) holds.
Hence it remains to prove that I 2 ÝÑ 0 as N Ñ 8 for any fixed R ą 0.
Fix R ą 0 and recall that ϕ is a smooth function on R`satisfying ½ r0,1s ≤ ϕ ≤ ½ r0,2s . Define Hence it suffices to show that J 1 ÝÑ 0 as N Ñ 8 for each fixed R, m.
Note that H 1 R,m is bounded and jointly continuous in pµ, r, vq. Hence F 1 R,m is continuous and bounded on PpDpR`; R 2d qqq. Using the weak convergence π pN q ÝÑ π p8q as N Ñ 8 we conclude that also J 1 ÝÑ 0 as N Ñ 8, for each fixed R, m.

Uniqueness for bounded coefficients
In this section we study uniqueness for the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q in the case where σ is bounded, i.e. γ " 0. The following is our main result in this case.
Theorem 4.1. Suppose that γ " 0. Then for each µ 0 P PpR 2d q there exists at most one solution to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q. In particular, there exists at most one weak solution to the mean-field SDE (1.9).
The proof of this theorem is deduced from the following considerations. Given any solution µ to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q, then by taking expectations in (1.8) we see that its time-marginals pµ t q t≥0 satisfy the nonlinear Fokker-Planck equation xg, µ t y " xg, µ 0 y`t ż 0 xApµ s qg, µ s y, t ≥ 0, g P C 1 c pR 2d q, (4.1) where Apµ s q was defined in (1.6). Then we prove uniqueness for (4.1). Based on this uniqueness result, it suffices to study the corresponding linearized martingale problem where pµ t q t≥0 appearing in the argument of Apµ t q can be regarded as a fixed parameter. Uniqueness for the latter (time-inhomogeneous) martingale problem follows classically by uniqueness of its timemarginals.

Uniqueness for the time-marginals
In this section we study uniqueness and stability for the time-marginals, i.e. solutions to (4.1). More precisely, we prove an a priori bound for any two solutions to (4.1) with respect to the total variation distance where BpR 2d q denotes the space of all bounded measurable functions on R 2d . The proof of such bound relies on a mild formulation of (4.1) described below. Moreover, (4.2) naturally extends to all g P BpR 2d q.
The following is our main estimate for solutions to (4.1).
Taking the supremum over all g P BpR 2d q with }g} 8 ≤ 1 and then applying the Gronwall lemma yields the assertion.

Uniqueness in law for the Vlasov-McKean equation
Below we prove that the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q has at most one solution. Proposition 4.4. Suppose that γ " 0 and let µ 0 P PpR 2d q. Then there exists at most one solution µ P PpDpR`; R 2d qq to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q. Proof. Let µ and r µ be two solutions to the nonlinear martingale problem pA, C 1 c pR 2d q, µ 0 q. Their time-marginals pµ t q t≥0 and pr µ t q t≥0 both solve (4.1) and hence coincide, i.e. µ t " r µ t , for all t ≥ 0. Consequently gpxptqq´gpxp0qq´t ż 0 pApµ s qgqpxpsqqds, t ≥ 0 is a martingale with respect to µ and r µ, for any g P C 1 c pR 2d q. From this we readily conclude that µ " r µ, provided there exists at most one solution pρ t q t≥0 to the time-inhomogeneous Fokker-Planck equation xg, ρ t y " xg, ρ 0 y`t ż 0 xApµ s qg, ρ s y, t ≥ 0, g P C 1 c pR 2d q, apply e.g. [EK86, p.184, Theorem 4.2]. Uniqueness for pρ t q t≥0 can be shown in exactly the same way as Theorem 4.3.

Further uniqueness for unbounded coefficients
In this section we provide some sufficient condition for uniqueness and stability of solutions to (4.1) in the case where γ P p0, 2s.
Note that the additional integrability condition imposed on pµ t q t≥0 guarantees that xApµ s qg, µ s y in (4.1) makes sense. As before, it is not difficult to see that any solution to (4.1) still satisfies the mild formulation (4.2).

Estimate on the total variation distance
e δxvy γ µ t pdr, dvq. (5.1) The following is the main result on uniqueness and stability for (4.1).
Proceeding in the same way we can show that xASpt´sqg, pµ s´νs q b ν s y ≤ CC γ pδ, µ s`νs q}ν s } γ }µ s´νs } TV p1`| lnp}µ s´νs } TV q|q, which proves the assertion after taking the supremum over all g P BpR 2d q with }g} 8 ≤ 1. Uniqueness and stability is a direct consequence of the a priori estimate we have shown, i.e. one may apply a generalization of the Gronwall inequality stated in the appendix.

Estimate on the Wasserstein distance
In this part we prove estimates for solutions to (4.1) with respect to the Wasserstein distance dpµ, νq " sup where µ, ν P PpR 2d q are supposed to have finite first moments. Since particles are transported by the transport operator v¨∇ r , it is more natural to use the shifted Wasserstein distance where Sptqgpr, vq " gpr`vt, vq and Sptq˚is the adjoint operator defined by the relation xSptqg, µy " xg, Sptq˚µy, g P BpR 2d q, µ P PpR 2d q.
Below we will use another characterization of the shifted distance in terms of optimal couplings described as follows. Introduce a one-paramter family of metrics on R 2d |pr, vq´pr r, r vq| t :" |pr´vtq´pr r´r vtq|`|v´r v|, t ≥ 0 and related to this metrics define the time-dependent Lipschitz norms }g} t " sup pr,vq‰pr r,r vq |gpr, vq´gpr r, r vq| |pr, vq´pr r, r vq| t .
Note that this norms are all equivalent due to 1 1`t |pr, vq´pr r, r vq| t ≤ |pr, vq´pr r, r vq| 0 ≤ p1`tq|pr, vq´pr r, r vq| t .
Given µ, ν P PpR 2d q, a coupling H of pµ, νq is a probability measure on R 4d such that its marginals are given by µ and ν, respectively, i.e. for all g 1 , g 2 P C b pR 2d q one has ż R 4d pg 1 pr, vq`g 2 pr r, r vqq dHpr, v; r r, r vq " xg 1 , µy`xg 2 , νy.
Let Hpµ, νq the space of all such couplings. The reader may consult [Vil09] for additional details on couplings and related Wasserstein distance.
Proposition 5.3. Let µ, ν P PR 2d q satisfy ş R 2d p|r|`|v|qpµ`νqpdr, dvq ă 8 and fix t ≥ 0. Then there exists H t P Hpµ, νq such that |pr, vq´pr r, r vq| t dH t pr, v; r r, r vq. (5.2) Proof. The first equality follows from the definition of Sptq˚, the second equality from the definition of the norms }¨} t while the third equality is a particular case of the Kantorovichduality (see [Vil09]).
The following is our main coupling estimate for the Wasserstein distance d t .
For t ≥ 0, let H t P Hpµ t , ν t q be such that |pr, vq´pr r, r vq| t dH t pr, v; r r, r vq.
Then there exists CpT, a, ψq ą 0 (independent of µ t , ν t ) such that, for any t ≥ 0, Λpr, v, q, w; r r, r v, r q, r wqdH 0 s dH 1 s ds where dH 0 s " dH s pr, v; r r, r vq, dH 1 s " dH s pq, w; r q, r wq and Λpr, v, q, w; r r, r v, r q, r wq " pxvy`xwy`xr vy`x r wyq|σpv´wqψpr´qq´σpr v´r wqψpr r´r qq| p|pr, wq´pr r, r wq| s`| pr, vq´pr r, r vq| s q mintσpv´wq, σpr v´r wqu Proof. It is not difficult to see that both solutions still satisfy the mild formulation (4.2) for any g with }g} 0 ≤ 1. Hence we obtain rpASp´sqgqpr, v; r r, r vq´pASp´sqgqpq, w; r q, r wqs dH 0 s dH 1 s ds ": I.
Using }Sp´sqg} s ≤ 1 we obtain t|pr, w`uq´pr, vq| s`| pr r, r w`uq´pr r, r vq| s uˇˇψσ´r ψr σˇˇapuqdudH 0 s dH 1 s ds p|w`u´v|`| r w`u´r v|qˇˇψσ´r ψr σˇˇapuqdudH 0 s dH 1 s ds pxvy`xwy`xr vy`x r wyqˇˇψσ´r ψr σˇˇdH 0 s dH 1 s ds where we have used |w`u´v|`| r w`u´r v| ≤ Cxuypxvy`xwy`xr vy`x r wyq in the last inequality. Using again }Sp´sqg} s ≤ 1 gives Sp´sqgpr, w`uq´Sp´sqgpr r, r w`uq ≤ |pr, w`uq´pr r, r w`uq| s " |pr, wq´pr r, r wq| s , Sp´sqgpr r, r vq´Sp´sqgpr, vq ≤ |pr, vq´pr r, r vq| s .
Hence J 1 is estimated by p|pr, wq´pr r, r wq| s`| pr, vq´pr r, r vq| s q pψσ^r ψr σqapuqdudH 0 s dH 1 s ds p|pr, wq´pr r, r wq| s`| pr, vq´pr r, r vq| s q pσ^r σqdH 0 s dH 1 s ds which proves the assertion.
The following gives the main estimate for this section.
Proof. It is easily seen that the general coupling inequality is applicable in this case. Let us start with the first term in Λ. Using the elementary inequality c a,b |x a`b´ya`b | ≤ px a`ya q|x b´yb | ≤ C a,b |x a`b´ya`b |, x, y ≥ 0, a, b ą 0 we obtain |σpv´wqψpr´qq´σpr v´r wqψpr r´r qq| ≤ σpv´wq |ψpr´qq´ψpr r´r qq|`ψpr r´r qq |σpv´wq´σpr v´r wq| ≤ C pxvy γ`x wy γ q p|r´r r|`|q´r q|q`C p|v´r v|`|w´r w|q and hence pxvy`xwy`xr vy`x r wyq|σpv´wqψpr´qq´σpr v´r wqψpr r´r qq| ≤ C`xvy 1`γ`x wy 1`γ`x r vy 1`γ`x r wy 1`γ˘p |r´r r|`|q´r q|`|v´r v|`|w´r w|q ≤ C`xwy 1`γ`x r wy 1`γ˘p |r´r r|`|v´r v|q`C`xvy 1`γ`x r vy 1`γ˘p |q´r q|`|w´r w|q C`xvy 1`γ`x r vy 1`γ˘p |r´r r|`|v´r v|q`C`xwy 1`γ`x r wy 1`γ˘p |q´r q|`|w´r w|q .
Hence using that H 0 s , H 1 s P Hpµ s , ν s q we obtain ż R 8d pxvy`xwy`xr vy`x r wyq|σpv´wqψpr´qq´σpr v´r wqψpr r´r qq|dH 0 s dH 1 p|r´r r|`|v´r v|q dH s pr, v; r r, r vq C ż R 4d`x vy 1`γ`x r vy 1`γ˘p |r´r r|`|v´r v|q dH s pr, v; r r, r vq vy 1`γ`x r vy 1`γ˘| pr, vq´pr r, r vq| s dH s pr, v; r r, r vq ≤ CC γ pT, µ`ν, δqd s pµ s , ν s qp1`| lnpd s pµ s , ν s qq|qd s pµ s , ν s q, where we have used |r´r r|`|v´r v| ≤ p1`T q|pr, vq´pr r, r vq| s , (5.3) and similar arguments to the proof of Theorem 5.2 (see also [FRS18b] and [FM09]) to obtain ż R 4d`x vy 1`γ`x r vy 1`γ˘| pr, vq´pr r, r vq| s dH s pr, v; r r, r vq ≤ CC γ pT, µ`ν, δqd s pµ s , ν s qp1`| lnpd s pµ s , ν s qq|q.
Applying the general coupling inequality and then above estimates proves the assertion.
Remark 5.6. Using again Lemma 6.1 from the Appendix we may deduce from above estimate uniqueness and stability with respect to the Wasserstein metric.
6 Appendix 6.1 Proof of Lemma 1.3 (a) Applying the Itô formula we obtain, for g P C 1 c pR 2d q, where pM g ptqq t≥0 is a local martingale. It suffices to show that pM g ptqq t≥0 is, indeed, a martingale. For each g P C 1 c pR 2d q we find C ą 0 with xwy γ dµ s pq, wqxvy γ " C}µ s } γ xvy γ .

This implies that
Ep sup sPr0,ts (b) Let pq t , w t q be a measurable process defined on pr0, 1s, Bpr0, 1sq, dηq such that pq t , w t q has law µ t , for all t ≥ 0, where µ t denotes the time-marginal of µ. Using [HK90, Theorem A.1] gives the existence of a weak solution pR, V q to (1.9) such that pR, V q has law µ.

Proof of Lemma 2.1
By the mean-value Theorem we get |v j`u | 2p " p|v j | 2`| u| 2`2 v j¨u q p " p|v j | 2`| u| 2 q p`2 pp|v j | 2`| u| 2 q p´1 pv j¨u q 4ppp´1qpv j¨u q 2 1 ż 0 p1´tq`|v j | 2`| u| 2`2 tpv j¨u q˘p´2 dt For the last integral we get by 2|v j ||u| ≤ |v j | 2`| u| 2 and pa`bq q ≤ 2 q pa q`bq q for q ≥ 0 and a, b ≥ 0ˇˇˇˇˇˇ4 Let k p " t p`1 2 u where txu P Z is defined by txu ≤ x ă txu`1, set`p l˘" ppp´1q¨¨¨pp´l´1q l! , for l ≥ 1, and`p 0˘" 1. Then we obtain by the fractional binomial expansion (see e.g. [LM12, Lemma 3.1]) where we have used k p ≤ p. Using the symmetry of a we have ş R d pv j¨u qapuqdu " 0 and hence obtain ż where we have used ř kp l"1`p l˘≤ 2 p ≤ 2 3p´2 and ppp´1q ≤ 2 p to obtain By symmetry we obtain N ÿ k,j"1 ψpr k´rj qσpv k´vj q`|v j | 2p´| v k | 2p˘" 0 (6.1) and hence Since k p ≤ p´1 we obtain from the Young inequality xv k y γ xv j y 2kp ≤ xv k y γ xv j y 2p´2 ≤ γ 2p´2`γ xv k y 2p´2`γ`2 p´2 2p´2`γ xv j y 2p´2`γ .
Next by 2k p`γ ≤ 2p´2`γ we obtain xv j y 2kp`γ ≤ xv j y 2p´2`γ . Putting all estimates together we deduce the assertion.

Some variants of the Gronwall lemma
We need the following generalization of the Gronwall inequality (see [Che95, Lemma 5.2.1, p. 89]) for a proof).
Lemma 6.1. Let ρ be a nonnegative bounded function on r0, T s, a P r0, 8q and g be a strictly positive and non-decreasing function on p0, 8q. Suppose that ş 1 0 dx gpxq " 8 and ρptq ≤ a`t ż 0 gpρpsqqds, t P r0, T s.

Then
(a) If a " 0, then ρptq " 0 for all t P r0, T s.
(b) If a ą 0, then Gpaq´Gpρptqq ≤ t where Gpxq " ş 1 x dy gpyq . The following nonlinear generalization of the Gronwall lemma is a particular case of the Bihari-LaSalle inequality.

Some localization result
Let pE, ρq be a complete, separable metric space. Let A Ă C b pEqˆCpEq be a (multi-valued) operator such that there exists 1 ≤ ψ P CpEq with |g| ≤ K f ψ, @pf, gq P A (6.2) for some K f ą 0. Set P ψ :" µ P PpEq | ş E ψpxqdµpxq ă 8 ( . Here and below DpR`; Eq denotes the Skorokhod space and x the canonical process on DpR`; Eq. Definition 6.3. Let µ P P ψ . A solution to the martingale problem pA, µq is a probability measure P µ on DpR`; Eq such that (a) P µ pxp0q P Aq " µpAq for all A P BpEq. is a martingale w.r.t. P µ .
When working with martingale problems the use of localization techniques such as [EK86, Theorem 6.3, Corollary 6.4] is essential. However, the statements therein require that A Ă C b pEqˆBpEq, i.e. ψ " 1. Below we give one possible extension.
Theorem 6.4. Let A Ă C b pEqˆCpEq satisfy (6.2) and A m Ă C b pEqˆCpEq be such that |g m | ≤ K f ψ holds for pf, g m q P A m with a constant K f ą 0 independent of m ≥ 1. Suppose that there exists µ P P ψ such that the following conditions hold: (i) There exist open sets pU m q m≥1 with U m Ă U m`1 , Ť m≥1 U m " E and tpf, ½ Um gq | pf, gq P A m u " tpf, ½ Um gq | pf, gq P Au , m ≥ 1.
Moreover ½ Um ψ is bounded for any m ≥ 1.
(ii) The martingale problem pA m , ρq has for each ρ P PpEq and each m ≥ 1 a unique solution.
(iii) We have lim kÑ8 sup m≥k P m µ pτ k ≤ T q " 0, @T ą 0 where P m µ is the unique solution to the martingale problem pA m , µq and τ k " inftt ą 0 | xptq R U k or xpt´q R U k u is a stopping time on DpR`; Eq.
(iv) There exists p ą 1 such that for all T ą 0 there exists Cpp, T q ą 0 satisfying where E m µ denotes the expectation w.r.t. P m µ . Then there exists a unique solution P µ to the martingale problem pA, µq. This solution satisfies sup tPr0,T s E µ pψpxptqq p q ≤ Cpp, T q, T ą 0.
Remark 6.5. In several cases one may take U m " tx P E | ψpxq ă mu. In such a case condition (iii) is implied by lim Proof.
Take H with }H} BL ≤ 1, T ą 0, 1 ≤ k ă m and set x T :" xp¨^T q, H T pxq :" Hpx T q. Then |E m µ pHq´E k µ pHq| ≤ |E m µ pH T q´E m µ pHq|`|E m µ pH T q´E k µ pH T q|`|E k µ pH T q´E k µ pHq| ": I 1`I2`I3 .

Then by
Step 1 and ½ τmąT ≥ ½ τ k ąT we get I 2 ≤ }H} 8´P m µ pτ m ≤ T q`P k µ pτ k ≤ T q¯`}H} 8 E m µ p½ τmąT´½τ k ąT q " }H} 8´P m µ pτ m ≤ T q`P k µ pτ k ≤ T q¯`}H} 8´P m µ pτ m ą T q´P k µ pτ k ą T qw hich tends by (iii) clearly to zero. Moreover we have qpxpt^u^T q, xpt^uqqdu‚≤ e´T and likewise I 3 ≤ e´T which completes Step 2.
Step 3. Let P µ be the limit of P m µ . Using (iv), monotone convergence and the Lemma of Fatou one can show that sup tPr0,T s E µ pψpxptqq p q ≤ Cpp, T q, T ą 0.
Step 4. Take g P CpEq such that there exists K g ą 0 with |g| ≤ K g ψ. We show that lim mÑ8 E m µ pgpxptqqq " E µ pgpxptqqq , t P D µ where D µ " tt ≥ 0 | P µ pxptq " xpt´qq " 1u. Note that D c µ is at most countable.