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Distribution dependent SDEs for Landau type equations

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  • Wang, Feng-Yu

Abstract

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup Pt∗ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup Pt∗ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.

Suggested Citation

  • Wang, Feng-Yu, 2018. "Distribution dependent SDEs for Landau type equations," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 595-621.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:2:p:595-621
    DOI: 10.1016/j.spa.2017.05.006
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    References listed on IDEAS

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    1. Arnaudon, Marc & Thalmaier, Anton & Wang, Feng-Yu, 2009. "Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3653-3670, October.
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    Cited by:

    1. Ning, Ning & Wu, Jing & Zheng, Jinwei, 2024. "One-dimensional McKean–Vlasov stochastic variational inequalities and coupled BSDEs with locally Hölder noise coefficients," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
    2. Ren, Panpan & Wu, Jiang-Lun, 2021. "Least squares estimation for path-distribution dependent stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    3. Jianhai Bao & Xing Huang, 2022. "Approximations of McKean–Vlasov Stochastic Differential Equations with Irregular Coefficients," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1187-1215, June.
    4. Ma, Xiaocui & Yue, Haitao & Xi, Fubao, 2022. "The averaging method for doubly perturbed distribution dependent SDEs," Statistics & Probability Letters, Elsevier, vol. 189(C).
    5. Yifan Bai & Xing Huang, 2023. "Log-Harnack Inequality and Exponential Ergodicity for Distribution Dependent Chan–Karolyi–Longstaff–Sanders and Vasicek Models," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1902-1921, September.
    6. Fan, Xiliang & Yu, Ting & Yuan, Chenggui, 2023. "Asymptotic behaviors for distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 383-415.
    7. Huang, Xing & Wang, Feng-Yu, 2019. "Distribution dependent SDEs with singular coefficients," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4747-4770.
    8. Ren, Panpan, 2023. "Singular McKean–Vlasov SDEs: Well-posedness, regularities and Wang’s Harnack inequality," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 291-311.
    9. Fan, Xiliang & Huang, Xing & Suo, Yongqiang & Yuan, Chenggui, 2022. "Distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 23-67.
    10. Yulin Song, 2020. "Gradient Estimates and Exponential Ergodicity for Mean-Field SDEs with Jumps," Journal of Theoretical Probability, Springer, vol. 33(1), pages 201-238, March.
    11. Sharrock, Louis & Kantas, Nikolas & Parpas, Panos & Pavliotis, Grigorios A., 2023. "Online parameter estimation for the McKean–Vlasov stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 481-546.
    12. Li, Butong & Meng, Yongna & Sun, Xiaobin & Yang, Ting, 2022. "Optimal strong convergence rate for a class of McKean–Vlasov SDEs with fast oscillating perturbation," Statistics & Probability Letters, Elsevier, vol. 191(C).

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