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Elliptic Equations Associated with Brownian Motion with Singular Drift

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Abstract

In this paper, we first obtain the existence and uniqueness of solution u of elliptic equation associated with Brownian motion with singular drift. We then use the regularity of the weak solution u and the Zvonkin-type transformation to show that there is a unique weak solution to a stochastic differential equation when the drift is a measurable function.

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References

  1. Bass, R.F., Chen, Z.Q.: Brownian motion with singular drift. Ann. Probab. 31, 791–817 (2003)

    Article  MathSciNet  Google Scholar 

  2. Chen, Z.Q., Wang, L.M.: Uniqueness of stable processes with drift. Proc. Am. Math. Soc. 144(6), 2661–2675 (2016)

    Article  MathSciNet  Google Scholar 

  3. Kim, P., Song, R.: Two sided estimates on the density of Brownian motion with singular drift. Ill J. Math. 50(1–4), 635–688 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Krylov, N.V., Roeckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131(2), 154–196 (2005)

    Article  MathSciNet  Google Scholar 

  5. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, vol. 24. Elsevier, Amsterdam (2014)

    MATH  Google Scholar 

  6. Yang, S., Zhang, T.S.: Dirichlet boundary value problems for elliptic operators with measure data: a probabilistic approach. arXiv preprint arXiv:1804.01819 (2018)

  7. Zhang, T.: A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients. Ann. Probab. 39(4), 1502–1527 (2011)

    Article  MathSciNet  Google Scholar 

  8. Zhang, X.: Strong solutions of SDEs with singular drift and Sobolev diffusion coefficients. Stochast. Process. Appl. 115(11), 1805–1818 (2005)

    Article  MathSciNet  Google Scholar 

  9. Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR-Sbornik 22(1), 129 (1974)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the referees for their valuable comments which helped to considerably improve the quality of the paper. This work is partly supported by National Natural Science Foundation of China (No.11671372, No.11721101, No.11971456) and the Fundamental Research Funds for the Central Universities, China (No. WK0010000057).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Saisai Yang

  2. School of Mathematics, University of Manchester, Oxford Road, Manchester, England, M13 9PL, UK

    Chen Wang & Tusheng Zhang

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  1. Saisai Yang
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  2. Chen Wang
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  3. Tusheng Zhang
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Correspondence to Saisai Yang.

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Yang, S., Wang, C. & Zhang, T. Elliptic Equations Associated with Brownian Motion with Singular Drift. Commun. Math. Stat. 10, 101–122 (2022). https://doi.org/10.1007/s40304-020-00213-8

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  • DOI: https://doi.org/10.1007/s40304-020-00213-8

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