Communications in Mathematics and Statistics

, Volume 6, Issue 4, pp 533–581 | Cite as

Singular Brownian Diffusion Processes

  • Xicheng ZhangEmail author
  • Guohuan Zhao


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.


Singular drift Weak solution Heat kernel Ergodicity Zvonkin’s transformation 

Mathematics Subject Classification

60H10 35A08 37A25 


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Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China
  2. 2.Applied MathematicsChinese Academy of ScienceBeijingPeople’s Republic of China

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