Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications

Abstract

In this paper, couplings by change of measure are constructed to derive log-Harnack inequalities for conditional McKean-Vlasov SDEs, where the diffusion coefficients corresponding to the common noise are distribution free or merely depend on the distribution variable and for the latter one, the stochastic Hamiltonian system is also considered. Moreover, the quantitative propagation of chaos in Wasserstein distance is obtained, which combined with the coupling by change of measure implies the quantitative propagation of chaos in the relative entropy and total variation distance.

Comparison of Difficulty Levels of Examples in High School Mathematics Textbooks

“…In contrast to McKean-Vlasov SDEs without common noise, the research on McKean-Vlasov SDEs with common noise is not too rich. Yet, in the past few years, there are still some progresses on qualitative and quantitative analyses; see, for example, [3,22,37] on well-posedness, and [6,16,23,37] concerned with finite-time conditional propagation of chaos. According to [6, p. 110-112], the random distribution flow (µ t ) t>0 associated with (1.3) solves the nonlinear FPE:…”

“…Concerning the SDE (2.1), in this subsection, we handle the phenomenon on conditional propagation of chaos in a finite-time horizon. In the past few years, this subject has achieved some progresses; see, for example, [6, Theorem 2.12] and [23,Theorem 2.3], where the drift term and the diffusion term are Lipschitz continuous with respect to the spatial variables, and [26, Proposition 2.1], in which the coefficients satisfy the monotone condition with respect to the spatial variables. It is worthy to emphasize that the coefficients of McKean-Vlasov SDEs with common noise under investigation in [6,26,23] are L 2 -Wasserstein Lipschitz continuous with respect to the measure variables.…”

“…In the past few years, this subject has achieved some progresses; see, for example, [6, Theorem 2.12] and [23,Theorem 2.3], where the drift term and the diffusion term are Lipschitz continuous with respect to the spatial variables, and [26, Proposition 2.1], in which the coefficients satisfy the monotone condition with respect to the spatial variables. It is worthy to emphasize that the coefficients of McKean-Vlasov SDEs with common noise under investigation in [6,26,23] are L 2 -Wasserstein Lipschitz continuous with respect to the measure variables. Yet, in the present paper, the drift parts of the conditional McKean-Vlasov SDEs we are interested in are L 1 -Wasserstein Lipschitz continuous.…”

“…Yet, in the present paper, the drift parts of the conditional McKean-Vlasov SDEs we are interested in are L 1 -Wasserstein Lipschitz continuous. In particular, the corresponding convergence rate of conditional propagation of chaos was revealed in [6,26,23] whenever the initial distributions enjoy high order moments. As far as we are concerned, the quantitative convergence rate of conditional propagation of chaos is unnecessary for our purpose, so the high order moment of the initial distribution is dispensable as showed in the following proposition.…”