A second order analysis of McKean–Vlasov semigroups

Uniform in time weak propagation of chaos on the torus

“…A similar result is also available in [60]. Lastly, in the recent contribution [2], the weak error (1.5) is shown to be O(N −1 + N −1/d exp(−λt)), for λ > 0, when the McKean-Vlasov dependence is small enough with respect to the confinement properties of the drift. The result is stated for test functions Φ that are linear in the measure argument.…”

“…Description of the three cases. Obviously, the condition imposed in (1) is reminiscent of those required in [33] and [2] on the smallness of the mean field dependence. The three papers indeed share a common idea: the threshold for the intensity of the mean field dependence is determined in terms of the ergodic properties of the (linear) Fokker-Planck equation obtained by 'removing' the mean field term in (1.4).…”

“…The key fact is that κ > 0 may be large, in which case (3) is known as the synchronised (or super-critical) Kuramoto model. Reformulated with the notations of example (2), this corresponds to W (x) = −2πκ cos(2πx). Obviously, it is not H-stable.…”

“…When specialising to a linear function Φ(µ) = T d F (x)µ(dx), this says that F has to be merely Hölder continuous. In contrast, F is required to be twice differentiable (with bounded derivatives) in [2]. Our strategy of allowing such weaker conditions combines two main ingredients.…”

“…Even more, although we restrict ourselves here to the periodic case as we already find it very rich, we feel that our method could be applied to the Euclidean setting, provided that a suitable form of confining drift is added to the dynamics. In this regard, it is worth mentioning that the approach used in [2] to handle the Euclidean case is of a somewhat different essence (in addition to the fact that the class of test functions is not the same). Therein, the linearisation procedure is mostly achieved at the level of the McKean-Vlasov equation itself.…”