On a strong form of propagation of chaos for McKean-Vlasov equations

By Daniel Lacker. 2018

117 citations recorded by the source corpus.

This note shows how to considerably strengthen the usual mode of convergence of an $n$-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no interaction term. Notably, the empirical measure converges in a much stronger topology than weak convergence, and any fixed $k$ particles converge in total variation to their limit law as $n\rightarrow\infty$. This requires minimal continuity for the drift in both the space and measure variables. The proofs are purely probabilistic and rather short, relying on Girsanov's and Sanov's theorems. Along the way, some modest new existence and uniqueness results for McKean-Vlasov equations are derived.

Subfields: Finance, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2482ac3069022ee8880dd8fe57f232311a9e512a

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Citations in the corpus

Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance (2020, background): “Recently, the (weak and strong) well-posedness is studied in [3, 4, 6, 13, 17, 18, 20] for (1) with σt(x, γ) = σt(x) independent of the distribution variable γ, and with singular drift bt(x, γ).”
Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, background): “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.”
Simulation of McKean–Vlasov SDEs with super-linear growth (2018, background): “ee [6, Theorem 1.10] for example) lim N→∞ sup 1≤i≤N E sup 0≤t≤T |Xi,N t −X i t| 2 = 0. Several propagation of chaos results have been shown over the years under varying conditions, see [35], [29] and [23] among others. All SDEs appearing below have initial condition Xi 0 and we work on the interval ”
Well-posedness of distribution dependent SDEs with singular drifts (2018, background): “4), by the well-known result for bounded measurable drift b obtained in [22] (see also [18], [17] and [32]), for each n ∈ N, there is a solution to the following distribution dependent SDE: dX t = (∫”
Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions (2021, background): “As discussed above, prior work [34, 36, 37, 43, 49] has focused on establishing global entropy estimates like (1.”