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<p>This paper is a continuation of (Röckner and Zhao, Bernoulli <b>29</b>(1), 821 757–784 (2023)). Based on a compactness criterion for random fields in Wiener-Sobolev spaces, in this paper, we prove the strong solvability of time-inhomogeneous stochastic differential equations with drift coefficients in critical Lebesgue spaces, which gives an affirmative answer to a longstanding open problem. As an application, we also prove a regularity criterion for solutions of a stochastic system proposed by Constantin and Iyer (Comm. Pure. Appl. Math. 61(3): 330–345, 2008), which is closely related to the Navier-Stokes equations.</p>

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SDEs with critical time dependent drifts: strong solutions

  • 期刊论文

  • 发表日期：

    2025年6月19日

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  • Michael Röckner,
  • Guohuan Zhao

摘要

This paper is a continuation of (Röckner and Zhao, Bernoulli 29(1), 821 757–784 (2023)). Based on a compactness criterion for random fields in Wiener-Sobolev spaces, in this paper, we prove the strong solvability of time-inhomogeneous stochastic differential equations with drift coefficients in critical Lebesgue spaces, which gives an affirmative answer to a longstanding open problem. As an application, we also prove a regularity criterion for solutions of a stochastic system proposed by Constantin and Iyer (Comm. Pure. Appl. Math. 61(3): 330–345, 2008), which is closely related to the Navier-Stokes equations.