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<p>We prove the existence of probabilistically strong solutions for large classes of possibly degenerate stochastic differential equations with unbounded and locally Sobolev-regular coefficients, using the restricted Yamada–Watanabe theorem. Our approach relies on existence results for the corresponding Fokker–Planck equation, combined with both novel and existing restricted pathwise uniqueness results for SDEs. Here, restricted pathwise uniqueness means pathwise uniqueness among a subclass of weak solutions to the SDE. Furthermore, we derive new uniqueness results for the Fokker–Planck equation.</p>

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Strong solutions to degenerate SDEs and uniqueness for degenerate Fokker–Planck equations

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  • 发表日期：

    2025年10月10日

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  • Sebastian Grube

摘要

We prove the existence of probabilistically strong solutions for large classes of possibly degenerate stochastic differential equations with unbounded and locally Sobolev-regular coefficients, using the restricted Yamada–Watanabe theorem. Our approach relies on existence results for the corresponding Fokker–Planck equation, combined with both novel and existing restricted pathwise uniqueness results for SDEs. Here, restricted pathwise uniqueness means pathwise uniqueness among a subclass of weak solutions to the SDE. Furthermore, we derive new uniqueness results for the Fokker–Planck equation.