Singular Brownian Diffusion Processes

Xicheng Zhang , Guohuan Zhao

Communications in Mathematics and Statistics ›› 2018, Vol. 6 ›› Issue (4) : 533 -581.

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Communications in Mathematics and Statistics ›› 2018, Vol. 6 ›› Issue (4) : 533 -581. DOI: 10.1007/s40304-018-0164-7
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Singular Brownian Diffusion Processes

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Abstract

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.

Keywords

Singular drift / Weak solution / Heat kernel / Ergodicity / Zvonkin’s transformation

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Xicheng Zhang, Guohuan Zhao. Singular Brownian Diffusion Processes. Communications in Mathematics and Statistics, 2018, 6(4): 533-581 DOI:10.1007/s40304-018-0164-7

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Funding

NNSF(11731009)

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