论文标题
McKean-Vlasov方程的无限维正规化，并具有沃斯坦扩散
Infinite-dimensional regularization of McKean-Vlasov equation with a Wasserstein diffusion
论文作者
论文摘要
近年来，花了很多努力来恢复具有非平滑系数的McKean-Vlasov Sdes的唯一性。作为一个典型的实例，假定速度场在其空间变量和Lipschitz-colliniule中相对于其度量变量的总变化而言是有界和可测量的，请参见[Jourdain，Mishura-veretennikov]。与这些作品相反，我们在本文中考虑了由无限维噪声驱动的fokker-Planck方程，这是受到[Konarovskyi，Marx]研究的Wasserstein Space上的扩散模型的启发。我们证明，该方程的适合性能适用于漂移函数，只要有限维组成部分的规律性与度量参数中的规律性之间的规律性之间尊重，这可能只能在其度量参数中进行界限和测量。在这方面，我们表明，B相对于其空间变量的规律性越高，我们必须在B上对B的测量变量进行较低的规律性才能恢复唯一性。
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.
