Singular Brownian Diffusion Processes

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 solutio...

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Bibliographic Details
Published in:Communications in mathematics and statistics 2018-12, Vol.6 (4), p.533-581
Main Authors: Zhang, Xicheng, Zhao, Guohuan
Format: Article
Language:English
Subjects:
Brownian motion
Martingales
Mathematics
Mathematics and Statistics
Statistics
Well posed problems
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Summary: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.
ISSN:2194-6701
2194-671X
DOI:10.1007/s40304-018-0164-7