In this paper, we survey the recent progress about the SDEs with distributional drifts and generalize some well-known results about the Brownian motion with singular measure-valued drifts. In particular, we show the well-posedness of martingale problem or the existence and uniqueness of weak solutions, and obtain sharp two-sided and gradient estimates of the heat kernel associated with the above SDE. Moreover, we also study the ergodicity and global regularity of the invariant measures of the associated semigroup under some dissipative assumptions.

中文翻译：

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奇异布朗扩散过程

在本文中，我们调查了具有分布漂移的SDE的最新进展，并归纳了有关带有奇异量值漂移的布朗运动的一些著名结果。特别是，我们显示了problem问题的适定性或弱解的存在和唯一性，并获得了与上述SDE相关的热核的清晰的双边和梯度估计。此外，在某些耗散假设下，我们还研究了相关半群不变度量的遍历性和全局规律性。