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Weak Quantitative Propagation of Chaos Via Differential Calculus On Weak quantitative propagation of chaos via differential calculus on the space of measures Jean-François Chassagneux1, Lukasz Szpruch2, and Alvin Tse2 1LSPM, Université Paris Diderot 2School of Mathematics, University of Edinburgh Abstract d d Consider the metric space ( 2(R ), W2) of square integrable laws on R with the topology P d induced by the 2-Wasserstein distance W2. Let Φ: 2(R ) R be a function and µN be the empirical measure of a sample of N random variablesP distributed→ as µ. The main result of this paper is to show that under suitable regularity conditions, we have k−1 Cj 1 Φ(µ) EΦ(µN ) = + O( ), | − | N j N k j=1 X for some positive constants C1,...,Ck−1 that do not depend on N, where k corresponds to the degree of smoothness. We distinguish two cases: a) µN is the empirical measure of N-samples from µ; b) µ is a marginal law of McKean-Vlasov stochastic differential equation in which case µN is an empirical law of marginal laws of the corresponding particle system. The first case is studied using functional derivatives on the space of measures. The second case relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state the general regularity conditions required for each case and analyse the regularity in the case of functionals of the laws of McKean-Vlasov SDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties. arXiv:1901.02556v1 [math.PR] 8 Jan 2019 1 Introduction The aim of this work is to provide an exact weak error expansion between a (nonlinear) functional Φ : (Rd) R of the empirical measure µ (Rd) and its deterministic limit Φ(µ), µ (Rd). P2 → N ∈ P2 ∈ P2 We distinguish two cases: a) µN is the empirical measure of N-samples from µ; b) µ is the marginal law of a process described by a McKean-Vlasov stochastic differential equation (McKV-SDE), in which case µN is the empirical measure of the marginal laws of the corresponding particle system. 1 In the first case where µN is the empirical measure of N-samples from µ, the only interesting case is when the functional Φ is non-linear. To provide some context to our results, one may, for example, assume that Φ is Lipschitz continuous with respect to the Wasserstein distance, i.e, there exists a constant C > 0 such that Φ(µ) Φ(ν) CW (µ,ν), µ,ν (Rd) , | − |≤ 2 ∀ ∈ P2 one could bound Φ(µ) EΦ(µ ) by EW (µ,µ ). Consequently, following [15] or [14], the rate of | − N | 2 N convergence in the number of samples N deteriorates as the dimension d increases. On the other hand, recently, authors [13, Lem. 5.10] made a remarkable observation that if the functional Φ is twice-differentiable with respect to the functional derivative (see Section 2.1.1), then one can obtain a dimension-independent bound for the strong error E Φ(µ) Φ(µ ) p, p 4, which is of order | − N | ≤ O(N −1/2) (as expected by CLT). Here, we study a weak error and show that, (see Theorem 2.17) if Φ is (2k + 1)-times differentiable with respect to the functional derivative, then indeed we have k−1 C 1 Φ(µ) EΦ(µ ) = j + O( ), | − N | N j N k Xj=1 for some positive and explicit constants C1,...,Ck−1 that do not depend on N. The result is of inde- pendent interest, but is also needed to obtain a complete expansion for the error in particle approx- imations of McKV-SDEs that we discuss next. The second situation we treat in this work concerns estimates of propagation-of-chaos type 1. Consider a probability space (Ω, , P) with a d-dimensional Brownian motion W . We are interested F in the McKean-Vlasov process Xs,ξ with interacting kernels b and σ, starting from a random { t }t∈[s,T ] variable ξ, defined by the SDE2 t t X0,ξ = ξ + b(X0,ξ, L (X0,ξ)) dr + σ(Xs,ξ, L (X0,ξ)) dW , t [s, T ], (1.1) t r r r r r ∈ Z0 Z0 s,ξ s,ξ d d d where L (Xr ) denotes the law of Xr and functions b = (b ) : R (R ) R and σ = i 1≤i≤d × P2 → (σ ) : Rd (Rd) Rd Rd satisfy suitable conditions so that there exists a unique weak i,j 1≤i,j≤d × P2 → ⊗ solution (see e.g. [34] or, for more up-to-date panorama on research on existence and uniqueness, see [32, 20, 1, 12]). McKean-Vlasov SDE (1.1) can be derived as a limit of interacting diffusions. Indeed, 0,ξ N 1 N one can approximate the law L (X ) by the empirical measure X := δ j,N generated by N · · N j=1 X· particles i,N defined as {X }1≤i≤N P t t i,N = ξ + b i,N , XN dr + σ i,N , XN dW i, 1 i N, t [0, T ], (1.2) Xt i Xr r Xr r r ≤ ≤ ∈ Z0 Z0 1We would like to remark that our results also cover a situation where the law µ is induced by a system of stochastic differential equations with random initial conditions that are not of McKean-Vlasov type in which case the samples are i.i.d. 2We assume without loss of generality that the dimensions of X and W are the same because we will not make any non-degeneracy assumption on the diffusion coefficient σ in our work. In particular, one dimension of X could be time itself. 2 where W i, 1 i N, are independent d-dimensional Brownian motions and ξ , 1 i N, are i.i.d. ≤ ≤ i ≤ ≤ random variables with the same distribution as ξ. It is well known, [34, Prop. 2.1], that the property XN of propagation of chaos is equivalent to weak convergence of measure-valued random variables t L N L XN Rd to (Xt). A common strategy is to establish tightness of π = ( t ) ( ( )) and to identify N ∈ P P the limit by showing that π converges weakly to δL (Xt). This approach does not reveal quantitative bounds we seek in this paper, but is a very active area of research. We refer the reader to [18, 34, 29] for the classical results in this direction and to [22, 3, 16, 30, 26] for an account (non-exhaustive) of recent results. On the other hand, the results on quantitative propagation of chaos are few and far in between. In the case when coefficients of (1.1) depend on the measure component linearly, i.e., are of the form b(x,µ)= B(x,y) µ(dy), σ(x,µ)= Σ(x,y)µ(dy) , Rd Rd Z Z with B, Σ being Lipschitz continuous in both variables, it follows from a simple calculation [34] to see that W (L ( i,N ), L (X0,ξ)) = O(N −1/2). We refer to Sznitman’s result as strong propagation of 2 Xt t chaos. Note that in this work we treat the case of McKean-Vlasov SDEs with coefficients with general measure dependence. In that case, as explicitly demonstrated in [7, Ch. 1], the rate of strong propaga- tion of chaos deteriorates with the dimension d. This is due to the fact that one needs to estimate the difference between the empirical law of i.i.d. samples from µ and µ itself using results such as [15] or [14]. In the special case when the diffusion coefficient is constant, with linear measure dependence on the drift (which lies in some negative Sobolev space), the rate of convergence in the total variation norm has been shown to be O(1/√N) in [21]. Of course, in a strong setting, O(1/√N) is widely considered to be optimal as it corresponds to the size of stochastic fluctuations as predicted by the CLT. In this work, we are interested in weak quantitative estimates of propagation of chaos. Indeed, this new direction of research has been put forward very recently by two independent works [24, Ch. 9] and [31, Th. 2.1]. The authors presented novel weak estimates of propagation of chaos for linear functions in measure, i.e. Φ(µ) := F (x)µ(dx) with F : Rd R being smooth. This gives the rate Rd → of convergence O(1/N), plus the error due to approximation of the functional of the initial law (see R [31, Lem. 4.6] for a discussion of a dimensional-dependent case). While the aim of [31] is to estab- lish quantitative propagation of chaos for the Boltzmann’s equation, in a spirit of Kac’s programme [23, 28], Theorem 6.1 in [31, Th. 6.1] specialises their result to McKV-SDEs studied here, but only for elliptic diffusion coefficients that do not depend on measure and symmetric Lipschitz drifts with linear measure dependence. The key idea behind both results is to work with the semigroup that acts on the space of functions of measure, sometimes called the lifted semigroup, which can be viewed a dual to the space of probability measures on (Rd) as presented in [30].
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