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Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term

Author

Listed:
  • G. Prato

    (Scuola Normale Superiore)

  • F. Flandoli

    (Università di Pisa)

  • E. Priola

    (Università di Torino)

  • M. Röckner

    (Bielefeld University)

Abstract

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $$B$$ B and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato et al. in Ann Probab 41:3306–3344, 2013) which generalized Veretennikov’s fundamental result to infinite dimensions assuming boundedness of the drift term. As in Da Prato et al. (Ann Probab 41:3306–3344, 2013), pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift $$B$$ B is assumed only to be measurable and bounded and grow more than linearly.

Suggested Citation

  • G. Prato & F. Flandoli & E. Priola & M. Röckner, 2015. "Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1571-1600, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0545-0
    DOI: 10.1007/s10959-014-0545-0
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    References listed on IDEAS

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    1. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
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    Cited by:

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    2. David Criens, 2019. "Cylindrical Martingale Problems Associated with Lévy Generators," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1306-1359, September.
    3. Pappalettera, Umberto, 2022. "Large deviations for stochastic equations in Hilbert spaces with non-Lipschitz drift," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 1-20.
    4. David Criens & Moritz Ritter, 2022. "On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2052-2067, September.
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