Abstract
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.
Highlights
This note develops a simple but apparently new approach to analyzing McKean-Vlasov stochastic differential equations, of the form dXt = b(t, Xt, μt)dt + σ(t, Xt)dWt, μt = Law(Xt), ∀t ≥ 0, in which the drift is merely bounded and measurable, with fairly weak continuity requirements in the measure variable
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 → ∞
Propagation of chaos here refers to the convergence of the n-particle system, defined by the SDE
Summary
While several papers have studied McKean-Vlasov equations with discontinuities, the coefficients are often continuous enough, in the sense that the set of discontinuities has measure zero with respect to any candidate solution (see, e.g., [7]). In such a situation one can still apply the usual weak convergence arguments, which are not available for the general discontinuities in x we allow, more in the spirit of [14]. It is worth stressing that all of the proofs are purely probabilistic
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