{'fraction_non_alphanumeric': 0.07445796058209755, 'fraction_numerical': 0.02294474918239864, 'mean_word_length': 3.5570099353414286, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 34, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 28, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent

The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves the rate of convergence, and retains information about the fluctuations of stochastic gradient descent in the continuum limit.


INTRODUCTION
The analysis of machine learning algorithms is confronted with algorithms in high dimension, with a large number of degrees of freedom (parameters), huge data-sets, and high computing capacities. This motivates the analysis of scaling limits, corresponding to the asymptotic regimes in which these parameters become large, leading to a variety of relative scaling regimes. A particularly relevant one is the so-called overparametrised regime, which corresponds to the case in which the number of parameters M is much larger than the (large) number of training samples (data) N . Indeed, a large class of real-world algorithms fall into this class, see, for example [10].
The success of such algorithms comes as a statistical surprise. Classical belief and estimates in statistics would suggest that vast overparametrization leads to overfitting [83], contradicting empirical evidence in machine learning. A systematic explanation of this observation constitutes a key challenge in the scientific understanding of machine learning. For recent progress concerning the related "double-descent" phenomenon of the error observed in machine learning we refer to [8,40,65,74]. A central standing conjecture is that the (stochastic) learning algorithm employed in empirical risk minimization introduces an "implicit bias" towards minimizers that generalize well, thereby avoiding those that would lead to overfitting.
In order to prove or disprove this conjecture, universal models for machine learning are needed as the basis for the analysis of the stochastic dynamics of (stochastic) learning algorithms and their implicit bias. Motivated by this, several scaling limits of stochastic gradient descent dynamics have been analysed in the literature. In particular, the overparametrised regime, with its "mean-field" [15,45,78] and "lazy training" [14,44,84] scalings has achieved significant attention in recent years. In these works, the joint scaling limit of small learning rate α → 0 and overparametrization M → ∞ is considered. Roughly speaking, it is shown, for example in [66], that the empirical distribution ν M,α of the network parameters following stochastic gradient descent converges to the solution µ 0 of a (deterministic) gradient flow in the sense that This corresponds to a law of large numbers result, since it proves the concentration of the random measures ν M,α onto a deterministic path. The mean behavior of stochastic gradient descent can then be analysed by considering the limiting dynamics µ 0 . However, after passing to µ 0 all of the information about the inherent fluctuations of stochastic gradient descent is lost. Since there is substantial empirical evidence that stochasticity is decisive for the implicit bias of stochastic gradient descent [2,49,82], universal limiting models incorporating these fluctuations are needed.
In this work, we rigorously identify a class of nonlinear conservative SPDEs which serve as such a fluctuating continuum model. All previous known results rely on considering the joint scaling limit M → ∞, α → 0. In contrast, in practice, the sizes of networks M are typically large, while the learning rate α is moderately small (e.g. [81]). This corresponds to the scaling limit M → ∞ with α > 0 small but fixed. The identification of a scaling limit in this regime is demanding, since it informally corresponds to the solutions of a nonlinear SPDE with challenging well-posedness properties. The development of a corresponding well-posedness framework and the rigorous treatment of the scaling regime M → ∞, α > 0 are two of the main contributions of the present work.
We now give a more precise account of the setup and results of the present work. Supervised learning starts from a given training set of data 1 D ⊆ R n 0 × R k 0 with inputs Θ = {θ : (θ, f (θ)) ∈ D} and labels {f (θ) : (θ, f (θ)) ∈ D}. One then chooses a space of hypotheses. Here, we consider a fully-connected feed-forward network with one hidden layer with weights/parameters and φ a nonlinear activation function. In practice, φ is often chosen as the rectified linear unit (ReLU). In fact, the results of this paper apply to more general choices of Φ, and we only restrict for simplicity to the specific choice in this introduction. Thereby, we obtain a parametrization of the space of hypotheses M = {f M (x, θ) : x ∈ R M d }. The aim of risk minimization then is to select a suitable model f M (x, ·) ∈ M minimizing the risk where we concentrate on square loss l(x, y) = 1 2 |x − y| 2 , and ϑ is some measure on the data set D. For example, D finite with ϑ the uniform distribution corresponds to empirical risk minimization. This leads to the optimization problem Err := ! min x L(x). In machine learning, this optimization is approximated by variants of the stochastic gradient descent algorithm [73], corresponding to a random choice of the direction of descent. More precisely, the optimization dynamics are specified via (1.3) x(n + 1) = x(n) − α P P p=1 ∇ x l(f (x(n), θ p ), f (θ p )), 1 For simplicity we assume that the ground-truth is given by a function f .
where θ p are i.i.d. samples drawn from ϑ, α is the learning rate, P is the mini-batch size, and x(0) is initialized i.i.d from a measure µ 0 . Stochastic gradient descent corresponds to P = 1 and gradient descent to P = N . Note that, since E ϑ ∇ x l(f (·), f M (x, ·)) = ∇ x L(x), this is an unbiased estimator of gradient descent. The convergence of these (stochastic) optimization algorithms depends crucially on the properties of the empirical risk L. In machine learning, this risk landscape is typically non-convex, non-smooth, and degenerate, making the rigorous analysis of the convergence of stochastic gradient descent challenging. In a series of works, see, e.g. [30,61], it has been shown that for small learning rate α > 0, the time discrete dynamics (1.3) can be approximated up to first order by the following SDE (1. 5) and σ = α P is the fluctuation intensity. Notably, compared to plain gradient descent, (1.5) retains information on the fluctuations in (1.3), and offers a higher order of approximation.
In the case of shallow networks (1.1) and square loss (1.2) we observe, following [12,74,76], that we can represent the square loss as follows . This representation of the loss reveals intricate relations to statistical physics by interpreting the parameters as particles interacting via the interaction potential given by the risk (see [76] and A(x, µ) =Ã(x, x, µ). Based on this, in several works [12,74,76] it has been informally suggested that in the overparametrised limit (M → ∞) the dynamics will converge to solutions of the martingale problem where M is a continuous martingale satisfying (1.8) [ ψ, M α · ] t = t 0 ψ(x) ⊗ ψ(y) :Ã(x, y, µ α s )µ α s (dx)µ α s (dy)ds.
The proof of this conjecture is one of the main results of this work. Precisely, we prove that in the M → ∞ scaling limit, the empirical measure µ M,α converges to a solution µ α of the conservative SPDE (1.9), with the optimal rate of convergence The rigorous proof relies on the development of a well-posedness framework for (1.7), that is, to an infinite dimensional martingale problem with nonlocal coefficients and degenerate ellipticity. We approach this by analyzing instead the probabilistically strong well-posedness of SPDEs that have (1.7) as their martingale problem. However, the naive guess for such an SPDE, corresponding to informally taking the square-root of the coefficients in the quadratic variation (1.8), leads to an SPDE with irregular diffusion coefficients, for which the validity of strong uniqueness appears unclear. Instead, as one of the first main ideas of this work, we introduce an alternative "coupling" SPDE associated to (1.7), by (1.9) which we will call the Stochastic Mean-Field Equation, where W is a cylindrical Wiener process in L 2 (Θ, ϑ). We then prove the (probabilistically strong) well-posedness of this SPDE. Since (1.9) is an SPDE with nonlocal coefficients, and degenerate coercivity, this is a challenging task. We next state a paraphrased version of our main results on the well-posedness of (1.9). We refer the reader to Theorems 2.16, 2.23, and 2.26 for the precise results under varying assumptions on the coefficients and the initial data. . Given sufficiently nice initial data µ 0 ∈ P 2 (R d ) and sufficiently nice coefficients V (·), A(·), G(·, ·), the stochastic mean-field equation (1.9) has a unique superposition solution in the sense of Definition 2.6.
The proof of this result relies on establishing a superposition principle for the stochastic meanfield equation (1.9), i.e. the proof that each solution to (1.9) is given as a superposition of solutions to the SDE with interaction where µ t = µ 0 • X(·, t) −1 . While the superposition principle for deterministic PDE has been well-established in a series of ground-breaking works [3,5,23,34,79], even in infinite dimensions [71], the few existing results for the case of SPDEs [16,35] rely on restrictive assumptions on the regularity of the coefficients or initial data, which are not satisfied in the present case. Therefore, a new argument for the case of measure-valued, discrete initial data corresponding to (1.6) and Lipschitz continuous coefficients is developed in the present work, as well as a new proof for measure-valued initial data, relaxing the regularity assumptions on the coefficients from [16]. We then prove the uniqueness of solutions to this Lagrangian system, which by the superposition principle implies the uniqueness of solutions to (1.9). Based on the superposition principle, we next establish the convergence of the empirical measures µ M,α to the solution µ α of the stochastic mean-field equation (1.9) with optimal rate M − 1 2 .
where µ α is a superposition solution of the stochastic mean-field equation (1.9) with σ = α P and initial datum µ 0 , W p denotes the p-Wasserstein distance, and the implicit constants are independent of α.
We next analyze the law of large numbers behavior of solutions µ α to (1.9) in the limit of small learning rate α → 0, proving an optimal rate of convergence to the deterministic transport equation Theorem 1.3 (See Theorem 3.1). If µ α is a superposition solution of (1.9) with initial datum µ 0 , then in the limit α → 0, we have the estimate where µ 0 is the solution of the transport equation (1.11) with initial datum µ 0 and the implicit constant is independent of α > 0.
Combining Theorems 1.2 and 1.3, we conclude which implies that the limits M → ∞ and α → 0 of µ M,α commute. In addition, using the results of the present work the intermediate limits can be characterized: Indeed, since µ M,α is itself shown to be a superposition solution to the stochastic mean-field equation (1.9), (1.12) implies that taking the limit α → 0 for M > 0 fixed yields a solution µ M to the transport equation (1.11) with initial datum µ M 0 . Subsequently, considering the limit M → ∞ and applying (1.13) implies that µ M converges to a solution to the transport equation (1.11) with initial datum µ 0 . Taking the limits in the opposite order follows in an analogous manner, see Figure 1.
Having established the law of large numbers behavior of µ M,α , we next turn to its asymptotic fluctuations and prove a quantified central limit theorem for the stochastic mean-field equation (1.9), again providing optimal bounds on the rate of convergence. As discussed in [76], there are two sources of fluctuations, one due to the sampling from the initial measure µ 0 and one due to the dynamical fluctuations in SGD. We define the corresponding fluctuation field η M,α := min{α −1/2 , M 1/2 }(µ M,α − µ 0 ).
As is done in [76,Section 4.4], we will focus on the case where the dynamical fluctuations dominate, i.e. α decreases slowly than M −1 . Precisely, assuming that M = M (α) with lim α→0 α −1 M −1 (α) < ∞, we prove that the fluctuation field η α = α −1/2 (µ M (α),α − µ 0 ) for α → 0 converges to a solution of the linear SPDE Overview of the literature. To first order, in the small learning rate limit α → 0, stochastic gradient descent converges to deterministic gradient descent. As argued above, this law of large numbers scaling limit does not incorporate information on the fluctuations of stochastic gradient descent. However, considering higher order approximations, stochastic gradient descent can be shown to converge to solutions to so-called stochastic modified equations with Σ given in terms of the variance of the stochastic sampling of the empirical loss, see [30,60]. For the validity of this limit for moderately large learning rates see [62]. A discussion of (1.16) with jump noise can be found in [67]. The effect of the randomness inherent to stochastic gradient descent on the implicit bias and on implicit regularization has been analyzed in [2,49,82].
Overparameterised limits of shallow networks in the mean-field training regime have received considerable attention in recent years. In [15,45,66,75,78] the convergence of gradient descent to a Wasserstein gradient flow has been shown and analysed. Notably, this limit is different from the "lazy training" regime which can be treated in terms of a linearisation around initialization, see [29,44]. An instructive comparison of the scaling regimes is given in [14] and of their performance in [13,38,70].
Linear SPDE have been rigorously identified in the context of central limit fluctuations in stochastic gradient descent in [75,77]. A fluctuating, nonlinear mean-field limit, incorporating the fluctuations of stochastic gradient descent was, informally, suggested in [12,74,76], taking the form of the conservative SPDE (2.1) below. The rigorous derivation of this conservative SPDE as well as a proof of a quantified central limit theorem remained open problems in the literature. These are solved in the present work.
For an overview of the literature on conservative SPDEs we refer to [33, Section 1.1]. We here concentrate on nonlocal conservative SPDEs. In [20] nonlocal, nonlinear stochastic Fokker-Planck equations have been considered, proving the uniqueness of solutions by several methods, e.g. by duality arguments, coupling arguments, and the Krylov-Rozovskii variational framework. Under less restrictive assumptions on the coefficients and solutions, this has been extended in [58, 25, p. 115]. The case of measure-valued solutions has been treated in [16]. Additionally, motivated by applications to fluid dynamics, signed measure-valued solutions to nonlocal, nonlinear stochastic Fokker-Planck equations have been considered in the literature, see, for example, e.g. [6,55,56,72] and the references therein. The convergence of interacting particle systems to solutions of nonlocal, nonlinear Fokker-Planck equations, and the closely related phenomenon of propagation of chaos has been considered in [16], [54] and [59,Theorem 2.3] and the references therein. SDE with interaction have been analyzed in [9,25,26,26,28,68], and their relation to SPDEs has been considered in [27,54].
The SPDE considered in this work bears some similarity with the so-called Dean-Kawasaki equation introduced in [22,48], and which corresponds to (1.6) for independent particles. In contrast to the Dean-Kawasaki equation, the noise caused by SGD is spatially correlated, which allows the development of a full mathematical treatment introduced in the present work. The more singular case of the Dean-Kawasaki equation has attracted considerable interest in the literature, yielding the construction of (renormalized) solutions [7,50,53,80], negative results on the existence of non-trivial solutions [51,52], and regularized models [18,19].
Central limit theorems for conservative, local SPDEs have been analyzed in [24]. The case of linear transport noise has been analyzed in [36]. For central limit theorems for parabolic SPDEs with multiplicative, semilinear noise we refer to [11,41,42] and the references therein. Higher order approximations of interacting particle systems by conservative, local SPDEs have been shown in [24], and for non-interacting particle systems up to arbitrary order in [17]. The authors are not aware of any previous results on central limit theorems for nonlocal conservative SPDEs.

1.2.
Outline of the paper. In Section 2, well-posedness results for the SDE with interaction (1.10) is shown assuming Lipschitz continuity of its coefficients, and the existence of a superposition solution to the stochastic mean-field equation (1.9) is established. This allows us to connect the uniqueness with the superposition principle in Corollary 2.12. The well-posedness of (1.10) and the continuous dependence of its solutions with respect to the initial particle distribution is obtained in Section 2.1. Section 2.2 is devoted to the proof of the uniqueness for the stochastic mean-field equation (1.9). The limit Theorems 3.1 and 3.7 are proved in Section 3. In Section 4, the higher order approximation of the SGD dynamics by solutions to the stochastic mean-field equation is obtained.
1.3. Basic notation. Let d ∈ N be fixed. For m ∈ N 0 := N ∪ {0} the space of m-times continuously differentiable functions from an open set Γ ⊂ R d to R is denoted by C m (Γ). The subspace of C m (Γ) of all bounded together with their derivatives (resp. compactly supported) functions is denoted by C m b (Γ) (resp. C m c (Γ)). We write C(Γ), C b (Γ) and C c (Γ) for C 0 (Γ), C 0 b (Γ) and C 0 c (Γ), respectively. Let ϕ, f i ∈ C m (Γ), i ∈ [n] := {1, . . . , n}. We set For vectors a, b ∈ R d and matrices A, B ∈ R d×d we will use the notation , where f = (f i ) i∈[n] and g = (g i ) i∈ [d] .
For p ≥ 0 we also introduce the subset C 2 For every m ∈ N 0 and δ ∈ (0, 1) denote by C m,δ (R d ) the subset of all functions from C m (R d ) whose m-th derivatives are locally δ-Hölder continuous, that is, a function f ∈ C m (R d ) belongs to C m+δ (R d ) if for every R > 0 there exists a constant C such that for all x, y ∈ R d with |x − y| ≤ R and α ∈ N d 0 with |α| = m. We equip the space C m,δ (R d ) with the Fréchet topology generated by the following seminorms We also denote byC m,δ the set of all functions f ∈ C m (R d × R d ) whose mixed m-th derivatives are locally δ-Hölder continuous, that is, for every R > 0 there exists a constant C > 0 such that , the spacẽ C m,δ (R d ) will be equipped with the Fréchet topology generated by the seminorms be the Hilbert space of all 2-integrable functions on R d with respect to the Lebesgue measure with the usual L 2 -norm · L 2 and inner product ·, · L 2 .
For J ∈ N 0 and an open domain Γ ⊂ R d we denote the complete extension of the space C ∞ c (Γ) with respect to the norm defined by by H J (Γ). It is well-known that H J (Γ) is a separable Hilbert space with the inner product The dual space to H J (Γ) equipped with the norm . It is also a separable Hilbert space with the inner product denoted by ·, · −J,Γ . According to the Riesz representation theorem, there exists the isometry between H J (Γ) and H −J (Γ) denoted by L J,Γ . We will often drop Γ from the notation of the inner product and the norm on a Sobolev space, if it does not lead to the confusion. For more details about the Sobolev spaces H J (Γ) and H −J (Γ) we refer the reader to, e.g., [1].
For I = [0, T ] or I = [0, ∞) the space of all càdlàg functions from I to a metric space E equipped with the Skorohod topology will be denoted by D(I, E). The set C(I, E) of all continuous functions from I to E is a closed subset of D(I, E) and the induced topology on C(I, E) is equivalent to the topology of uniform convergence on compacts.
The space of all probability measures (resp. signed measures with finite total variations) on (R d , B(R d )) equipped with the topology of weak convergence will be denoted by P(R d ) (resp. by M(R d )). Let P p (R d ) denote the subset of P(R d ) of all probability measures with finite p-moment for p ≥ 1, that is, where φ p (x) = |x| p , x ∈ R d , and ϕ, µ (and also ϕ(x), µ(dx) ) is the integration of ϕ with respect to µ. It is well-know that P p (R d ) is a Polish space with the Wasserstein distance given for each µ, ν ∈ P p (R d ) by where the infimum is taken over all probability measures on R d × R d with marginals µ and ν.
We will fix a measure space (Θ, G, ϑ) such that ϑ is a finite measure and the space L 2 (Θ, ϑ) := L 2 (Θ, G, ϑ), which consists of all 2-integrable with respect to ϑ functions (more precisely, equivalence classes) from Θ to R, is separable. The usual inner product and norm on L 2 (Θ, ϑ) are denoted by ·, · ϑ and · ϑ , respectively. Let W t , t ≥ 0, be a cylindrical Wiener process on L 2 (Θ, ϑ) defined on a complete probability space (Ω, F, P) and (F t ) t≥0 be the right-continuous complete extension of the filtration generated by W , which exists according to [46,Lemma 7.8]. We recall that for an (F t )-progressively measurable L 2 (Θ, ϑ)-valued process g t , t ≥ 0, satisfying where G t = g t , · ϑ , t ≥ 0, is an (F t )-progressively measurable process on the space of Hilbert-Schmidt operators on L 2 (Θ, ϑ). Denote the space of all Hilbert-Schmidt operators from L 2 (Θ, m) to a Hilber space H by L 2 (L 2 (Θ, m); H) and · HS,H be the Hilbert-Schmidt norm on that space. In particular, G t HS,R = g t ϑ .

WELL-POSEDNESS AND SUPERPOSITION PRINCIPLE
In this section, we establish the well-posedness and the superposition principle for the stochastic mean-field equation where W is a cylindrical Wiener process in L 2 (Θ, ϑ) defined on a complete probability space (Ω, F, P) and the functions V : × Ω → R d×d satisfy the following Assumption 2.1.
holds for every t ≥ 0.
All integrals in the definition above are well-defined due to the a.s. boundedness of the functions inside the integral ·, µ s . Remark 2.3. If for every T > 0 and compact set K in P 2 (R d ) the coefficients V and G are a.s. bounded on [0, T ] × R d × K, that is, there exists a (random) constant C > 0 such that and µ t , t ≥ 0, is a solution to (2.1), then the integral equality (2.2) in Definition 2.2 holds for every ϕ ∈ C 2 b (R d ). This follows from the dominated convergence theorem and the compactness of {µ t , t ∈ [0, T ]} in P 2 (R d ) due to the continuity of µ t , t ≥ 0.
Remark 2.4. Let µ t , t ≥ 0, be a solution to the equation (2.1). Then for every ϕ ∈ C 2 c (R d ) the process ϕ, µ t , t ≥ 0, is a continuous local (F t )-semimartingale with quadratic variation The expression for the quadratic variation of ϕ, µ t , t ≥ 0, directly follows from (1.17).
Together with the stochastic mean-field equation (2.1), we will consider the following associated SDE with interaction s. for all t ≥ 0 and for every u ∈ R d a.s.
for all t ≥ 0.
We remark thatμ t , t ≥ 0, is an (F t )-progressively measurable process in P 2 (R d ). Moreover, due to Fubini's theorem it does not depend on the version of X(u, ·), u ∈ R d , that is, if Y (u, t), satisfies the same measurablity conditions from Definition 2.5 as X and for every u ∈ R d a.s.
The key tool in the investigation of solutions to the stochastic mean-field equation (2.1) is the fact that it satisfies the (strong) superposition principle.
In order to build a strong superposition solution to the equation (2.1), we will need the Lipschitz continuity assumption on the coefficients, which will guarantee the well-posedness of the SDE (2.3).
Assumption 2.7. The coefficients V and G are Lipschitz continuous with respect to x and µ, that is, for every T > 0 there exists L > 0 such that a.s. for every t ∈ [0, T ], x, y ∈ R d and µ, ν ∈ P 2 (R d ) and Remark 2.8. Assumption 2 implies linear growth of V and G, that is, a.s.
for all x ∈ R d and µ ∈ P 2 (R d ). 3 The notion of the strong superposiltion solution considered in this work is close to one introduced by Flandoli (see [35,Definition 5]). Since we will only work with equation (2.3) which has a unique solution, we avoid more general definitions, like in [3,35,79], which needs the introduction of distributions on the path space.
In the next section, using a standard approach, we will prove the well-posedness of the SDE with interaction (2.3) stated in the following theorem.

Theorem 2.9 (Well-posedness of the SDE with interaction). Under Assumptions 2.1 and 2.7, the SDE with interaction (2.3) has a unique strong solution
The corollary directly follows from the inequality in Theorem 2.9 by its integration with respect to µ 0 .
Next, using Itô's formula, one can prove the existence of a superposition solution to (2.1).
Theorem 2.11 (Existence of solutions). Let V, G, A satisfy Assumptions 2.1 and 2.7. Then for every µ 0 ∈ P 2 (R d ) there exists a solution µ t , t ≥ 0, to the stochastic mean-field equation (2.1) started from µ 0 that satisfies the superposition principle and is moment preserving.
The proof of this theorem directly follows from Lemma 2.13 below. We note that the uniqueness of the stochastic mean-field equation is closely related to the superposition principle. Indeed, the well-posedness of the SDE with interaction (see Theorem 2.9) and Theorem 2.11 immediately imply the following corollary. In Section 2.2, we will prove the uniqueness to the stochastic mean-field equation making further assumptions on the initial condition and its coefficients.
2.1. SDE with interaction. We start this section with the proof that any solution to the SDE with interaction (2.3) provides a solution to the equation (2.1) in the sense of Definition 2.2.
Indeed, according to (1.17) and the polarisation equality, one gets Next, taking ϕ ∈ C 2 c (R d ) and applying Itô's formula to ϕ(X(u, t)) for every u ∈ R d , we get a.s.
Using the definition of the stochastic integral with respect to a cylindrical Wiener process, it is easily to see that Since ϕ has a compact support and the set x ∈ R d are bounded a.s., by Assumption 2.1. Hence, we may integrate the above expression with respect to µ 0 and use Fubini's theorem and the equality R d ψ(X(u, t))µ 0 (du) = ψ,μ t for ψ ∈ C b (R d ) to get thatμ t , t ≥ 0, satisfies (2.2). This ends the proof of the lemma. We next prove Theorem 2.9. Since its proof is similar to the proofs of [26, Theorem 2.2.1], we will only provide a sketch.
Proof of Theorem 2.9. Letμ 0 t = µ 0 , t ≥ 0. We define inductively for every n ≥ 1 the family of continuous processes {X n (u, t), t ≥ 0}, u ∈ R d , as solutions to the usual SDEs , t ≥ 0. Using Assumptions 2.1 and 2.7, it is easy to see that SDE (2.4) has a unique solution for every u ∈ R d and for every p ≥ 2, n ≥ 1 and T > 0 there exists a constant C > 0, independent of u, such that for all n ≥ 1 and u ∈ R d .
We will first show that X n satisfies the measurability assumptions from Definition 2.5. For this we will prove that X n has a continuous version in u. Taking u, v ∈ R d , T > 0, p ≥ 2, and using Hölder's inequality and the Burkholder-Davis-Gundy inequality, we estimate for each t where C 1 is a constant that depends only on p, T and d. By Assumption 2.7 and Gronwall's lemma, we get where C also depends only on p, T, L and d. Therefore, we can conclude from [46, Theorem 3.23] that X n (u, ·), u ∈ R d , has a continuous version as a C([0, ∞))-valued process, which will be also denoted by X n . Note that the choice of different version of X n (u, ·), u ∈ R d , does not change the fact that X n (u, ·) solves equation (2.4) for every u ∈ R d . Thus, the desired measurability of X n follows from the continuity of (u, t) → X n (u, t) a.s. We also note thatμ n , defined by (2.5), is an (F t )-progressively measurable continuous process in P 2 (R d ), where the continuity follows from bound (2.6) and de la Vallée-Poussin [63, Theorem 1.8]. Note thatμ · does not depend on the choice of a version of X n (u, ·), u ∈ R d . Next, using Assumption 2.7 again, we can estimate for every where C is independent of u, t and n. By Gronwall's lemma, we have Using the definition of the Wasserstein distance, we further estimate for t ∈ [0, T ] and n ≥ 2 where we have used the equality µ 0 (R d ) = 1 and (2.8) in the last step. Iterating the above inequality n − 2 times, we get The finiteness of the expectation on the right hand side of the above inequality follows from (2.6).
Next, using the Borel-Cantelli lemma, it is easily seen that there exist continuous processesμ t , as n → ∞. Moreover, for every u ∈ R d a.s.
for all t ≥ 0, by Assumption 2.7. Since the constant in inequality (2.7) does not depend on u, v and n, the inequality remains true for X n replaced by X, by Fatou's lemma. Therefore, the C([0, ∞))valued random field X(u, ·), u ∈ R d , has a continuous version, which is also denoted by X. This implies that X satisfies the measurability assumptions of Definition 2.5. We also remark that for every ϕ ∈ C b (R d ) and t ≥ 0 one has a.s.
by the dominated convergence theorem. This completes the proof of the well-posedness of SDE (2.3). We next show the finiteness of moments of the solution X. For each u ∈ R d and n, m ∈ N we define the stopping times Let also τ n,m u = τ n u ∧ σ m . Then, using Hölder's inequality, the Burkholder-Davis-Gundy inequality (see, e.g., Theorem 3.28 [47]) and Remark 2.8, we estimate for p ≥ 2 and every n, m ∈ N, where the constant C depends only on L, p, d and T . By Gronwall's lemma, we get Making n → ∞ and using Fatou's lemma, we obtain In order to bound the integral in the inequality above, we first estimate for all t ∈ [0, T ] and m ≥ 1. Combining the inequality above with (2.9), we obtain where the constant C depends only on L, p, d and T . Now, making m → ∞ and using Fatou's lemma, we complete the proof of the theorem.
We will further prove the continuous dependence ofμ · on the initial condition. This is known for SDEs with interaction with non-random coefficients driven by a Brownian sheet (see [ Then, for every T > 0, there exists a constant C > 0 such that Proof. Using Hölder's inequality, the Burkholder-Davis-Gundy inequality and Assumption 2.7, where C is independent of u, v and t. By Gronwall's lemma, we get for each t ∈ [0, T ] We next take an arbitrary probability measure . Then the probability measure χ t defined by , for each t ∈ [0, T ]. Thus, by the estimate above, we obtain Taking infimum over all χ with marginals where C depends only on T , L and d. Next, Gronwall's lemma implies This completes the proof of (2.11). The estimate (2.10) now directly follows from (2.11) and (2.12). This concludes the proof of the proposition.
We will also need to use the fact that two paths of particles described by an SDE with interaction never meet.
Lemma 2.15. Let V and G satisfy Assumptions 2.1, 2.7 and let X be a solution to (2.3). Then for Then by Itô's formula and Assumption 2.7, we can estimate for every T > 0 and t ∈ [0, T ] for some constant C independent of t and n. By Gronwall's lemma, E 1 |X(u,v,t∧σn)| < e CT |u−v| . Passing to the limit as n → ∞ and using Fatou's lemma, one gets s. for all t ≥ 0. Hence, σ = +∞ a.s. which completes the proof of the lemma.

2.2.
Uniqueness and superposition principle. The main goal of this section is to prove the uniqueness of solutions to the stochastic mean-field equation. We will consider separately a few types of initial particle distributions: atomic, with L 2 -density, and with a finite second moment. Depending on the type of initial conditions, we will need additional assumptions on the coefficients of the equation, which will appear in corresponding sections.
2.2.1. Atomic initial conditions. We remind the reader that φ m (x) = |x| m , x ∈ R d , and set It is easily seen that N n is a closed subspace of P 2 (R d ) for each n ≥ 1.
We define the following function F n : Remark 2.17. The function F n can be written as a polynomial of the maps To prove Theorem 2.16, we will apply Itô's formula to the semimartingale F (µ t ), t ≥ 0. Hence, we will need an analog of Itô's formula for functions of the form for some smooth functions f and ϕ l , l ∈ m, involving functional derivatives [21, Section 2] in order to simplify computations. We will often work with the linear stochastic mean-field equation obtained by freezing a solution µ t , t ≥ 0, in coefficients. In particular, we will write v(t, x), a(t, x) and g(t, x, θ) for V (t, x, µ t ), A(t, x, µ t ) and G(t, x, µ t , θ). ∇ϕ · g(s, ·, θ), ν s W (dθ, ds).

(2.17)
If there exist an (F t )-stopping time σ such that sup Proof. We will first show that equality (2.17) holds for every ϕ ∈ C 2 p (R d ) and t replaced by t ∧ σ. We fix g ∈ C 2 b (R) such that g(x) = 0, x ≤ 0, and g(x) = 1, x ≥ 1. Set for n ≥ 1 It is easy to see that for each x ∈ R d κ n (x) → 1 as n → ∞ and κ n , n ≥ 1, belongs uniformly to for all x ∈ R d and n ≥ 1. Let ϕ ∈ C 2 p (R d ) and ϕ n = ϕκ n , n ≥ 1. Note that ϕ n ∈ C 2 c (R d ) for all n ≥ 1. Therefore, for every t ≥ 0 ∇ϕ n · g(s, ·, θ), ν s W (dθ, ds).
We will next pass to the limit as n → ∞. A simple computation gives that ϕ n (x) → ϕ(x), Since the set {µ ∈ P 2 (R d ) : φ p , µ ≤ k} is closed in P 2 (R d ) and ν t , t ≥ 0, is a continuous process, σ k is an (F t )-stopping time, by [32, Proposition 2.1.5 (a)]. Let k ≥ 1 and t ≥ 0 be fixed. Since sup as n → ∞. By Remark 2.8, for every x ∈ R d and s ∈ [0, t ∧ σ k ] one has where C 1 is a constant which is independent of s and x. Since as n → ∞. Similarly, by Remark 2.8, for every s ∈ [0, t ∧ σ k ] where C 1 is also a constant independent of x and s. Therefore, using the dominated convergence theorem again, we get t∧σ k 0 D 2 ϕ n : a(x, ·), ν s ds → t∧σ k 0 D 2 ϕ : a(x, ·), ν s ds a.s. as n → ∞. It remains only to show the convergence of the stochastic integrals. We consider as n → ∞. In the last step, we have used the dominated convergence theorem, since |∇ϕ − ∇ϕ n | 2 |g(s, x)| 2 with a (non-random) constant C that does not depend on s and x, where the estimate can be obtained in a similar manner to (2.19). Consequently, Summarizing obtained convergence results, we can conclude that for each t ≥ 0 and k ≥ 0 we have a.s.
Using the equalities and we get (2.18) that completes the proof of the proposition.
Proof. We note that µ t , t ≥ 0, is a continuous process in P 2 (R d ). Hence, sup for all t ≥ 0. Consequently, the corollary directly follows from Proposition 2.18 with f (x) = x, x ∈ R, and the inclusion The following lemma gives the key property of solutions to the stochastic mean-field equation started from atomic initial condition that allows to prove their uniqueness.
Proof. To prove the lemma, it is enough to show that a.s. F n (ν t∧σ ) = 0, t ≥ 0, by observation (2.15). For this we will apply Itô's formula to the function F := F n , which satisfies assumptions of Proposition 2.18, according to Remark 2.17. We first compute the derivatives of F which appear in (2.18). For x ∈ R d we have where δ kl = I {k} (l) denotes the Kronecker delta. Similarly, for x, y ∈ R d and k, l ∈ [d] We next estimate the terms which appear after applying Itô's formula from Proposition 2.18 to the semimartingale F (ν t ), t ≥ 0. For the first term, we get Interchanging i 1 and i 2 , the expression above can be rewritten as Therefore, using the Lipschitz continuity of v(s, ·) from Assumption 2.7, we can estimate In order to estimate we first split J k,l 1 into three terms J k,l 1,1 , J k,l 1,2 , J k,l 1,3 defined by J k,l 1,1 (x, y, µ) = 4 which are obtained by summing over {i 3 : i 3 = i 2 , i 4 }, {i 3 : i 3 = i 2 , i 3 = i 4 } and {i 3 : i 3 = i 4 , i 3 = i 2 } in the second sum, respectively. We remark that J k,l 1,1 only appears for n ≥ 4 and J k,l 1,2 , J k,l 1,3 appear for n ≥ 3. We rewrite K as the sum K 1 + K 2 + K 3 , where In the above equality the functions b andb are defined by b k (s, ·) = a k,k (s, ·) andb k (s, ·) = a k,k (s, ·), k ∈ [d]. We next compute Therefore, using the Cauchy-Schwarz inequality and then the Lipschitz continuity of g(s, ·), we estimate We rewrite K 2 in similar way as K 1 : Interchanging the indexes of summations in second, third and fourth terms in the following way , respectively, and using the equalities a(s, x) = g(s, x, ·), g(s, x, ·) ϑ ,ã(s, x, y) = g(s, x, ·), g(s, y, ·) , we get − g(s, z i 1 , ·), g(s, z i 1 , ·) ϑ + g(s, z i 2 , ·), g(s, z i 1 , ·) ϑ − g(s, z i 2 , ·), g(s, z i 3 , ·) ϑ + g(s, z i 1 , ·), g(s, z i 3 , ·) ϑ Similarly as before, we get We now compute Hence, using the Lipschitz continuity of g(s, ·) again, we get

Combining obtained estimates for
, we can see that |K| ≤ C n L 2 F (ν s ). We next define for p = n 2 − n and every k ≥ 1 the (F t )-stopping time σ k as follows Then there exists a (non-random) constant C > 0 such that F (µ t ) ≤ C for all t ∈ [0, σ k ], by Remark 2.17. Applying Itô's formula to F (µ t ∧ σ k ), t ≥ 0, taking the expectations and using estimates obtained above, we have for every k ≥ 1 By Gronwall's lemma, E [F (µ t∧σ k )] = 0 for all t ≥ 0. This implies that for every t ≥ 0 a.s. µ t∧σ k ∈ N n−1 . Since N n−1 is closed and µ t , t ≥ 0, is continuous, we obtain that a.s. µ t∧σ k ∈ N n−1 for all t ≥ 0 and k ≥ 1. Making k → ∞, we can conclude that with probability 1 µ t∧σ ∈ N n−1 , t ≥ 0, that ends the proof of the proposition.
We next prove the main statement of this section.
Proof of Theorem 2.16. Let µ t , t ≥ 0, be a solution to the stochastic mean-field equation (2.1) started from µ 0 ∈ N n and satisfying (2.13). We will show that µ t is a superposition solution. Let . Without loss of generality, we may assume that α i , i ∈ [n], are strictly positive and x i , i ∈ [n], are distinct. Applying Lemma 2.21, we can conclude that a.s. µ t ∈ N n for all t ≥ 0. By the continuity of µ · , there exist (F t )-adapted continuous processes For every i ∈ [n] we define the stopping time Then, using the definition of a solution to the stochastic mean-field equation, we obtain since ∇ϕ i (x) = 0 and D 2 ϕ i (x) = 0 for |x − x i | < ε and for |x − x i | > 2ε. Similarly, using additionally the previous observation, for every t ≥ 0 holds. Hence, Y i is a solution to the usual SDE Note that (2.20) has at most one solution due to the Lipschitz continuity of its coefficients. Let X(u, t), t ≥ 0, u ∈ R d , be a solution to SDE with interaction (2.3). Since Next, considering the process µ 1 t := µ t+τ 0 conditioning to the σ-algebra F τ 0 , we get that, µ 1 t , t ≥ 0, satisfies the same stochastic mean-field equation with the initial condition µ τ 0 . Applying our argument above again to µ 1 Repining our argument infinitely many times and using the uniform continuity of Y i , i ∈ [n], a.s. on any compact time interval, we get that α i (t) = α i and Y i (t) = X(x i , t), t ∈ [0, τ ], for Since X(x i , ·) and X(x j , ·) never meet for distinct x i , x j , by Lemma 2.15, we can conclude that τ = +∞. Consequently, µ t = µ 0 • X −1 (·, t), t ≥ 0. This ends the proof of the uniqueness of the superposition principle for the stochastic mean-field equation.

2.2.2.
Initial condition with L 2 -density. In this section, we adapt the method from [58] in order to prove the uniqueness for the stochastic mean-field equation (2.1) if the initial condition has an L 2 -density with respect to the Lebesgue measure and the coefficients are bounded a.s. We will need the following assumption. |V (t, x, µ)| + |G(t, x, µ, ·)| ϑ ≤ L holds.
Using the notation dµ dx for the density of a measure µ ∈ P(R d ) with respect to the Lebesgue measure, we state the main result of this section. dx ∈ L 2 (R d ). Then (2.1) has a unique solution µ t , t ≥ 0, started from µ 0 . Moreover, µ t , t ≥ 0, is a superposition solution which is absolutely continuous with respect to the Lebesgue measure with for all t ≥ 0 and some constant C > 0.
Proof. The proof of the theorem is similar to the proof of the uniqueness result for the nonlinear SPDE in [58,Section 3], which differs from the equation (2.1) by non-random and homogeneous (independent of time) coefficients, on one side, and a more general structure on the other side. Therefore, we just describe the main steps omitting details. Let µ t , t ≥ 0, be a solution to (2.1) started from µ 0 . As in the previous section, we freeze µ t in the coefficients. As before, we set v(t, x) = V (t, x, µ t ), a(t, x) = A(t, x, µ t ) and g(t, x, θ) = G(t, x, µ t , θ). We will consider µ t , t ≥ 0, as a solution to the following linear SPDE This means that equality (2.17) holds for the process µ t , t ≥ 0, where test functions ϕ can be taken in C 2 b (R d ), by Corollary 2.20 and Assumption 2.7.
Our goal is to show that (2.22) has a unique solution. But now we will assume that a solution ν t , t ≥ 0, can take values in the space M(R d ) of all signed measures on R d with finite total variations and ν 0 ∈ M(R d ). Thus, let ν t , t ≥ 0, be a continuous (F t )-adapted process which satisfies (2.17) for all ϕ ∈ C b (R d ) and ν 0 is absolutely continuous with respect to the Lebesgue measure with dν 0 dx from L 2 (R d ). For any ρ ∈ M(R d ) and ε > 0, we define where p ε is the heat kernel defined by p ε (x) = 1 (2πε) −d/2 e − |x| 2 2ε . For every ϕ ∈ C b (R d ), we will also write P ε ϕ(x) for R d p ε (x − y)ϕ(y)dy and P ε (ϕρ)(x) for R d p ε (x − y)ϕ(y)ρ(dy). Setting ν ε t := P ε ν t , t ≥ 0, and following the computations from [58, Section 3], based on the application of Itô's formula, we get Using Assumptions 2.7, 2.22 and [58, Lemma 3.2], one can immediately conclude that for every T > 0 there exists a constant C such that where |ν s | is the total variation of ν s . Using the integration by parts in the L 2 -norm, the sum of the last two integrals in the expansion of E ν ε t 2 L 2 can be rewritten as This leads to the same estimate (2.23), see the proof of [58, Lemma 3.3] for more details. Hence, for every T > 0, there exists a constant C > 0 such that Following proofs of Corollary 3.1 and Theorem 3.3 from [58], we can conclude that every solution to the linear SPDE (2.22) with L 2 -density is unique. Moreover, if its solution ν t , t ≥ 0, is a measure-valued process, then ν t is absolutely continuous with respect to the Lebesgue measure with density satisfying (2.21) for every t ≥ 0. This immediately implies that the process µ t , t ≥ 0, is a unique solution to (2.22) started from µ 0 and is absolutely continuous with respect to the Lebesgue measure that satisfies (2.21).
In order to show that µ t , t ≥ 0, is superposition solution, consider a solution Y (u, t), t ≥ 0, u ∈ R d , to the equation which exists and is unique due to Theorem 2.9, since its coefficients v, g satisfy Assumptions 2.1, 2.7. By Lemma 2.13,ν t , t ≥ 0, is a solution to the (2.22) started from µ 0 . The uniqueness result, stated above, yields thatν t = µ t , t ≥ 0. On the other hand, the unique solution X to (2.3) solves also (2.24). Hence, Y must coincide with X. This completes the proof of the superposition principle.
Then, the uniqueness for the stochastic mean-field equation (2.1) directly follows from Corollary 2.12.

Initial condition with finite second moment.
We will now obtain the well-posedness and superposition principle for the stochastic mean-field equation for general initial conditions. This case includes both types of initial conditions considered before, however, stronger assumptions on the regularity of the coefficients will be needed. Our main idea is to transform a solution to the stochastic mean-field equation by a smooth (in space) stochastic flow to get a solution to a continuity equation with random coefficients for which the superposition principle can be easily obtain, e.g., by a duality method. This will imply that the original equation has only superposition solutions which will yield the uniqueness result by Corollary 2.12. For the construction of the transformation flow, we will mainly use results from [57].
We first introduce the following assumption on the coefficients of the equation and formulate the main result of this section.
, and for every T > 0 and a compact set whereÃ was defined in Remark 2.4.
Remark 2.25. Let µ t , t ≥ 0, be an arbitrary continuous process in P 2 (R d ). Then Assumption 2.24 implies that for each i, j ∈ [d] the processes t 0 V i (s, ·, µ s )ds, t ≥ 0, and t 0Ã i,j (s, ·, µ s )ds, t ≥ 0, are a.s. continuous in C 1,δ (R d ) andC 3,δ (R d ), respectively. Indeed, due to the continuity of the process µ t , t ≥ 0, the set K µ T := {µ t : t ∈ [0, T ]} is compact in P 2 (R d ) for every T > 0. Then the direct computation shows that for every , that guarantees the continuity. In order to prove the theorem we will state a few auxiliary statements. Let µ t , t ≥ 0, be a solution to the stochastic mean-field equation (2.1) whose coefficients satisfy Assumptions 2.1, 2.7 and 2.24. As before, we will freeze µ t in the coefficients, setting v(t, x) = V (t, x, µ t ), a(t, x) = A(t, x, µ t ),ã(t, x, y) =Ã(t, x, y, µ t ) and g(t, x, θ) = G(t, x, µ t , θ). We consider the following field of local martingales Note that its local quadratic variation Moreover, for each α with |α| ≤ 3, D α M (x, t), t ≥ 0, x ∈ R d , is a family of continuous local martingales with quadratic variation for any x, y ∈ R d , i, j ∈ [d] and α, β with |α| ≤ 3, |β| ≤ 3, by [57, Theorem 3.1.3].
We consider the following family of linear stochastic transport equations written in integral form where the Stratonovich integral was defined in [57,Section 3.2]. Using the connection between the Itô and Stratonovich integrals (see [57, Theorem 3.2.5]), one gets In order to compute the quadratic variation on the right hand side of the above expression, we will use [57, Theorem 3.
Therefore, equality (2.25) can be rewritten in Itô's form as follows for any t ≥ 0, x ∈ R d and k ∈ [d].
We will next consider for every t ≥ 0 the following probability measure ρ t = µ t • ψ −1 (t, ·) on R d . It is easy to see that ρ t , t ≥ 0, is a continuous process in P(R d ). Let us show that this process (locally) satisfies a continuity equation with random coefficients.
Lemma 2.28. Let the coefficients of (2.1) satisfy Assumptions 2.1, 2.7 and 2.24. Then the measurevalued process ρ t , t ≥ 0, defined above, is a solution to the equation Proof. Let ϕ ∈ C 2 c (R d ). We note that ϕ, ρ t = ϕ • ψ(t, ·), µ t for every t ≥ 0, by the definition of the measure ρ t . Hence, we first apply Itô's formula to ϕ • ψ(t, x), t ≥ 0, for each x ∈ R d . One gets for t ≥ 0 Next, we consider a non-negative function κ ∈ C 2 for each x ∈ R d and ε > 0. Integrating δ ε x by µ t and denoting µ ε t (x) = δ ε x , µ t , we get for every x ∈ R d a.s.
for all t ≥ 0. Using now the expressions for ϕ • ψ(t, x) and µ ε t (x) and Itô's formula, we obtain for every x ∈ R d a.s.
for all t ≥ 0. Note that for every t ≥ 0 a.s. there exists a compact K ⊂ R d such that supp ϕ • ψ(s, ·) ∈ K for any s ∈ [0, t]. Indeed, one can take K = {ψ −1 (s, x) : s ∈ [0, t], x ∈ supp ϕ}, which is a compact set as the image of [0, t] × supp ϕ under the continuous map (s, x) → ψ −1 (s, x) (for the continuity of ψ −1 see Proposition 2.27). Therefore, we may integrate the above expression with respect to the Lebesgue measure dx over R d . Then taking ε → 0, we get Next using the equalities ∇ y δ ε x (y) = −∇ x δ ε x (y), D 2 y δ ε x (y) = D 2 x δ ε x (y), the integration by parts formula and Fubini's theorem, we can conclude that the sum of all terms consisting of the integrals t 0 (. . . )ds in the expression for R d ϕ • ψ(t, x)µ ε t (x)dx converges a.s. to Taking into account that the second and the third integrals are the same and that the last term can be rewritten as we obtain ψ(s, ·)), µ s ds.
This directly implies that the process ρ t , t ≥ 0, solves the equation (2.28).
Note that we cannot control the growth of the coefficient b in the PDE (2.28) in the spatial variable. Therefore, the known superposition principle, e.g. from [4,23,79], cannot be applied. However, we can construct a superposition solution to (2.28) and a solution to its dual equation precisely, and then use the duality principle to prove that only the superposition solution solves (2.28).
Let Y (u, t), u ∈ R d , t ≥ 0, be a unique solution to (2.3) with the frozen coefficients v and g. In particular, for every u ∈ R d a.s.
whereb and b were defined in Lemma 2.28. Let ϕ ∈ C 2 c (R d ). By the usual chain-rule, one obtains for each u ∈ R d a.s.
We next prove the uniqueness of solutions to (2.28). Let ρ t , t ≥ 0, be a solution to (2.28). For a fixed function ϕ ∈ C 2 which is a C 3,δ ′ (R d )-valued continuous process. Then similarly to the proof of [57, Lemma 6.1.1], one can show that for every x ∈ R d a.s.
Repeating now the computation from the proof of Lemma 2.28, we get a.s.
We now ready to prove the well-posedness of the stochastic mean-field equation for arbitrary initial conditions.
Using that ψ(t, ·) is a bijection, we conclude that µ t = µ 0 • Y −1 (·, t), t ≥ 0, a.s. Hence, Y is a solution to the SDE with interaction (2.3) due to the definition of Y and the fact that µ t = µ 0 • Y −1 (·, t) = µ t for all t ≥ 0. Moreover, Y is also the unique solution, according to Theorem 2.9. This yields that µ t , t ≥ 0, is a superposition solution to (2.1). This completes the proof of the theorem since the superposition principle implies the uniqueness of the stochastic mean-field equation, by Corollary 2.12.
where C is independent of u, v, t and ε. We note that Hence, by Gronwall's lemma, we conclude for all t ∈ [0, T ], u, v ∈ R d and ε > 0, where C depends only on L and T . Note that in the last step of the inequality, we have used Theorem 2.9. For fixed ε > 0, let χ be an arbitrary probability measure on R d × R d with marginals µ ε 0 and µ 0 0 . Let also χ s (B) = χ{(u, v) : (X ε (u, s), X 0 (v, s)) ∈ B}, B ∈ B(R d × R d ). Then, for t ∈ [0, T ], we have the following Taking the infimum over all probability measures χ with marginals µ ε 0 and µ 0 0 , we obtain Using Gronwall's lemma again, we get the required inequality. This completes the proof of the theorem.
Proof. We first remark that G maps H −J+1 (Γ n ) to H −J (Γ n ). Indeed, we have and compute where β < α means β ≤ α and |β| < |α|. Hence, (3.7) Using the fact thatf ∈ C ∞ c (Γ n ) and the integration by parts, the last integral in the equality above can be rewritten as follows (3.8) Thus, Young's inequality and the equalities (3.7), (3.8) yield for every f ∈ R and some constant depending only on d and J. We now prove the estimate (3.6) for This completes the proof of the lemma.
We now prove the well-posedness of the linear SPDE (3.4).
Let also µ t , t ∈ [0, T ], be a continuous curve in and v(t, x) =v(t, x)+ ṽ(t, x, ·), µ t . Then, there exists a unique (F t )-adapted continuous process in H −J (R d ) such that for every ϕ ∈ C ∞ c (R d ) we have a.s.
Since for every n ∈N the space V n := H −J+1 (Γ n ) is continuously and densely embedded into the Hilbert space H n := H −J (Γ n ), is a Gelfand triple, where V * n is the dual space of V n with respect to the inner product ·, · Hn = ·, · −J,Γn . In particular, V * n f, ϕ Vn = f, ϕ Hn , f ∈ H n , ϕ ∈ V n . Let L 2 (L 2 (Θ, ϑ); V n ) denote the space of all Hilbert-Schmidt operators from L 2 (Θ, ϑ) to V n and · HS,Vn be the Hilbert-Schmidt norm on that space. For the cylindrical Brownian motion W t , t ≥ 0, in L 2 (Θ, ϑ) and fixed n ∈N, we consider the following equation (3.12) dξ n t = U n (t, ξ n t )dt + B n (t)dW t , where U n =Ū n +Ũ n : [0, T ] × V n → H n ⊂ V * n and B n : [0, T ] → L 2 (L 2 (Θ, ϑ); V n ) are measurable functions defined by Ū n (t, f ), ϕ 0,Γn = ∇ϕ · v(t, ·), f 0,Γn , and B n (t)h, ϕ 0,Γn = h, ∇ϕ(x) · g(t, x, ·), µ t (dx) ϑ for all ϕ ∈ C ∞ c (Γ n ), f ∈ V n and h ∈ L 2 (Θ, ϑ), where we extend the function ϕ to the whole space R d by zero in the integrals with respect to µ t , the function κ n ∈ C ∞ c (R d ) has a compact support in Γ n and κ n = 1 on Γ n−1 for every n ∈ N, κ ∞ = 1 on R d , and sup n≥1 κ n C J −1 b < ∞. The fact that U n and B n (t)h take values in H n and V n , respectively, follows from the continuous embedding of the Sobolev space H −J (Γ n ) into C 1 b (Γ n ). To prove the well-posedness of the equation (3.12), we need to check the following conditions for all n ∈ N.
(H0) There exists an orthogonal set {e 1 , e 2 , . . . } in V n such that it constitutes an orthonormal basis of H n . (H1) (Hemicontinuity) The map s → V * n U n (t, f 1 + sf 2 ), f Vn is continuous on R for any f 1 , f 2 , f ∈ V n and t ∈ [0, T ]. (H2) (Monotonicity) There exists a constant C > 0 such that for all t ∈ [0, T ] and f 1 , f 2 ∈ V n . (H3) (One-side linear growth) For any k ∈ N the operator U n (t, ·) maps the set E k := span{e 1 , . . . , e k } into V n , and there exists a constant C > 0 such that ) for all f ∈ E k and some constant independent of k and t. (H4) (Growth) There exists a constant C > 0 such that The condition (H0) follows from the compact embedding of the space V n into H n . Indeed, due to the fact that the inner product ·, · Vn induces a closed quadratic form on H n , there exists a densely defined self-adjoint operator L : H n ⊃ D(L) → H n on H n such that f 1 , f 2 Vn = f 1 , Lf 2 Hn for all f 1 ∈ V n and f 2 ∈ D(L). By the compact embedding of V n into H n , the operator L has a discrete spectrum (see also [31, Section 2.1]) {λ k , k ≥ 1} with corresponding eigenbasis {e k , k ≥ 1} in H n . It is easy to see that {e k , k ≥ 1} is an orthogonal system in V n and e k ∈ H −J+2 (Γ n ) for every k ≥ 1.
The condition (H1) holds due to the linearity of U n (t, ·) for every t ∈ [0, T ].
In order to check the monotonicity of U n , we estimate for f ∈ V n and t ∈ [0, T ] where the constant C is independent of n. We remark that sup is finite according to (3.9). For every f ∈ V n ⊂ H −J (Γ n ) we definef = L −1 J,Γn f ∈ H J (Γ n ) and note that f , ϕ J,Γn = f, ϕ 0,Γn for all ϕ ∈ C ∞ c (Γ n ). By [1,Theorem 5.29], the extension by zero off on R d , which we also denotef , belongs to H J (R d ). We next estimate (3.13) By the continuous embedding of Thus, the linearity of U n (t, ·) and the estimates above imply the monotonicity of U n with the constant C independent of n. This establishes condition (H2).
The function U n (t, ·) maps E k = span{e 1 , . . . , e k } to V n due to the fact that e k ∈ H −J+2 (Γ n ) for all k ≥ 1. Moreover, Lemma 3.3 and a similar computation to (3.13) imply Vn for all f ∈ E k , where the constant C is independent of n and k. Thus, condition (H3) holds.
We separately estimate the norms ofŪ n andŨ n in H n as follows for all f ∈ V n and t ∈ [0, T ], by the continuous embedding of for all f ∈ V n and t ∈ [0, T ], where the constant C is independent of n due to the specific choice of κ n . To estimate the Hilbert-Schmidt norm of B n (t), we consider an orthonormal basis {h k , k ≥ 1} in L 2 (Θ, ϑ) and compute for every k ≥ 1 and ϕ ∈ C ∞ c (Γ n ) where g k (t, x) = g(t, x, ·), h k ϑ . Hence, by the continuous embedding of where the constant C is independent of n. Consequently, by Jensen's inequality and Fubini's theorem, We note that sup (3.9). Thus, condition (H4) holds true with a constant C that is independent of n. Now we can apply [64, Theorem 1.1], to obtain that, for every n ≥ 1 and ̺ ∈ H −J+1 ⊂ V n , there exists a continuous H n -valued (F t )-adapted process η n t , t ∈ [0, T ], which has a dt × Pversionη n from L 2 ([0, T ] × Ω, dt × P; V ) and P-a.s. Hn + E T 0 η n t 2 where the constant C is independent of n because all constants in the conditions (H1)-(H4) are independent of n. Since for every m ≥ n the relationships H m ⊂ H n , V m ⊂ H n , and · Hn ≤ · Hm , · Vn ≤ · Vm hold, we get for all m ≥ n. Hence the sequence {η m , m ≥ n} is relatively compact in the weak topologies of L 2 (H n ) := L 2 ([0, T ] × Ω, dt × P; H n ), and L 2 (V n ) := L 2 ([0, T ] × Ω, dt × P; V n ) for every n ≥ 1. Using a diagonal argument, there exists a subsequence {m k , k ≥ 1} such that η m k →η in the weak topologies of L 2 (H n ) and L 2 (V n ) for every n ≥ 1. In particular, for all n ≥ 1. Using Fatou's lemma and the fact that the sequence η t Vn , n ≥ 1, increases, we get This yields thatη ∈ L 2 ([0, T ] × Ω, dt × P; H −J+1 (R d )) and Next, let γ ∈ L ∞ ([0, T ] × Ω, dt × P; R) and ϕ ∈ C c (R d ). We take n ∈ N satisfying supp ϕ ⊂ Γ n and compute ∇ϕ · g(s, ·, θ), µ s W (dθ, ds)dt.
Using the convergence of η m k in the weak topology of L 2 (V n ), we get ∇ϕ · v(s, ·),η s 0,R d dsdt.
Proof. Since for each ε ≥ 0 the process µ ε t , t ≥ 0, is a superposition solution to the corresponding equation, µ ε t = µ ε 0 • X ε (t, ·), t ≥ 0, where X ε is a solution to the SDE with interaction (2.3) with G replaced by √ εG. Similarly to the proof of Theorem 3.1, we fix ε > 0 and consider an arbitrary probability measure χ on R d × R d with marginals µ ε 0 and µ 0 0 . Using Remark 3.2 and the continuous embedding of (ϕ(X ε (u, t)) − ϕ(X 0 (v, t))) 2 χ(du, dv) Taking infimum over all χ with marginals µ ε 0 and µ 0 0 , we obtain the needed estimate, which completes the proof of the lemma.
To prove the convergence of the fluctuation field η ε , we will need the following assumptions on the coefficients of the stochastic mean-field equation (3.1).