Weak quantitative propagation of chaos via differential calculus on the space of measures

By Jean-François Chassagneux, Lukasz Szpruch, Alvin Tse. The Annals of Applied Probability, 2019

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Consider the metric space $(\mathcal{P}_2(\mathbb{R}^d),W_2)$ of square integrable laws on $\mathbb{R}^d$ with the topology induced by the 2-Wasserstein distance $W_2$. Let $\Phi: \mathcal{P}_2( \mathbb{R}^d) \to \mathbb{R}$ be a function and $\mu_N$ be the empirical measure of a sample of $N$ random variables distributed as $\mu$. The main result of this paper is to show that under suitable regularity conditions, we have \[ |\Phi(\mu) - \mathbb{E}\Phi(\mu_N)|= \sum_{j=1}^{k-1}\frac{C_j}{N^j} + O(\frac{1}{N^k}), \] for some positive constants $C_1, \ldots, C_{k-1}$ that do not depend on $N$, where $k$ corresponds to the degree of smoothness. We distinguish two cases: a) $\mu_N$ is the empirical measure of $N$-samples from $\mu$; b) $\mu$ is a marginal law of McKean-Vlasov stochastic differential equation in which case $\mu_N$ 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 It\^{o}-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.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3619e04e7f714530aaeb029a31a0197b290c8e9f

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Citations in the corpus

Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “That is, the functions (t, x,m) 7→ E[F (XT , μT ) |Xt = x, μt = m] should satisfy a certain linear PDE on [0, T ] × R × P(Rd), provided F is sufficiently nice [25, 22], and regularity estimates on the solution are useful for quantitative propagation of chaos arguments [34, 21].”
Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games (2020, methodology): “Therefore, such equations are very natural, and they were successfully used in the literature in the context of mean field limits of large particle system (see for instance in [34, 14]).”