In this paper, the distribution dependent stochastic differential equation in a separable Hilbert space with a Dini continuous drift is investigated. The existence and uniqueness of weak and strong solutions are obtained. Moreover, some regularity results as well as gradient estimates and Wang’s log-Harnack inequality are derived for the associated semigroup. In addition, Wang’s Harnack inequality with power and shift Harnack inequality are also proved when the noise is additive. All of the results extend the ones in the distribution independent situation.

中文翻译：

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具有奇异漂移的分布相关SPDE的适定性和规则性

本文研究了具有迪尼连续漂移的可分离希尔伯特空间中与分布有关的随机微分方程。得到了弱解和强解的存在性和唯一性。此外，还为相关的半群导出了一些规律性结果以及梯度估计和Wang的log-Harnack不等式。此外，当噪声是可加的时，也证明了王的带幂次的Harnack不等式和移位Harnack不等式。所有结果都在独立于分布的情况下扩展了结果。