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

Zhongmin Qian () and Yuhan Yao ()
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Zhongmin Qian: University of Oxford
Yuhan Yao: University of Oxford

Partial Differential Equations and Applications, 2022, vol. 3, issue 1, 1-22

Abstract: 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.

Keywords: Aronson estimates; Cameron–Martin formula; Diffusion processes; McKean–Vlasov SDEs; Strong solution; Vorticity equation; 60H30; 35Q30; 35Q35; 76D03; 76D05; 76D17 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-021-00146-z

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