arXiv 2002.101572020-02-24math.PR

具有Wasserstein扩散的McKean-Vlasov方程的Infinite-dimensional正则化

Infinite-dimensional regularization of McKean-Vlasov equation with a Wasserstein diffusion

Victor Marx

摘要：

近年来，人们在恢复非光滑系数McKean-Vlasov方程的唯一性方面做了大量的工作。作为一个典型的例子，速度场被假定为有界的，在其空间变量中是可测的，并且在其测量变量中关于全变差的距离是Lipschitz连续的，参见[Mishura-Veretennikov]。与这些作品相比，我们认为在本文中的福克-普朗克方程驱动的无限维噪声，灵感来自于扩散模型的Wasserstein空间研究[Konarovskyi，马克思]。我们证明了该方程的适定性适用于一个漂移函数，它可能只在其测量参数中是有界和可测量的，只要在有限维分量的正则性和测量参数的正则性之间进行权衡。在这方面，我们表明，更高的正则性的B相对于其空间变量，较低的正则性，我们必须假设B相对于其测量变量，以恢复唯一性。

英文摘要：

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 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, see [Jourdain, 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 in [Konarovskyi, Marx]. We prove that well-posedness of that equation holds 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.

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