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McKean–Vlasov type stochastic differential equations arising from the random vortex method

Author

Listed:
  • Zhongmin Qian

    (University of Oxford
    Oxford Suzhou Centre for Advanced Research)

  • Yuhan Yao

    (University of Oxford)

Abstract

We study a class of McKean–Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean–Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot–Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics.

Suggested Citation

  • Zhongmin Qian & Yuhan Yao, 2022. "McKean–Vlasov type stochastic differential equations arising from the random vortex method," Partial Differential Equations and Applications, Springer, vol. 3(1), pages 1-22, February.
    • Handle: RePEc:spr:pardea:v:3:y:2022:i:1:d:10.1007_s42985-021-00146-z
    DOI: 10.1007/s42985-021-00146-z
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    References listed on IDEAS

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    1. Qian, Zhongmin & Zheng, Weian, 2004. "A representation formula for transition probability densities of diffusions and applications," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 57-76, May.
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