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Infinite-dimensional regularization of McKean-Vlasov equation with a Wasserstein diffusion. (English) Zbl 1487.60115

Summary: Much effort has been spent in recent years on restoring uniqueness of McKean-Vlasov SDEs with non-smooth coefficients. As a typical instance, the velocity field \(b\) is assumed to be bounded and measurable in its space variable and Lipschitz-continuous with respect to the distance in total variation in its measure variable, as shown e.g. in the works of Jourdain and Mishura-Veretennikov. In contrast with those works, we consider in this paper a Fokker-Planck equation driven by an infinite-dimensional noise, inspired by the diffusion models on the Wasserstein space studied by Konarovskyi and von Renesse. We prove via Girsanov’s Theorem that there exists a unique weak solution to that equation for a drift function that might be only bounded and measurable in its measure argument, provided that a trade-off is respected between the regularity in the finite-dimensional component and the regularity in the measure argument. In this regard, we show that the higher the regularity of \(b\) with respect to its space variable is, the lower regularity we have to assume on \(b\) with respect to its measure variable in order to restore uniqueness in a weak sense.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q83 Vlasov equations
60J60 Diffusion processes
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