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Martingale solutions and Markov selections for stochastic partial differential equations

Author

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  • Goldys, Benjamin
  • Röckner, Michael
  • Zhang, Xicheng

Abstract

We present a general framework for solving stochastic porous medium equations and stochastic Navier-Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691-708] and Flandoli-Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Related Fields 140 (2008) 407-458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.

Suggested Citation

  • Goldys, Benjamin & Röckner, Michael & Zhang, Xicheng, 2009. "Martingale solutions and Markov selections for stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1725-1764, May.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:5:p:1725-1764
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    Cited by:

    1. Romito, Marco & Xu, Lihu, 2011. "Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 673-700, April.
    2. Zhu, Jiahui & Brzeźniak, Zdzisław & Liu, Wei, 2019. "Lp-solutions for stochastic Navier–Stokes equations with jump noise," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.
    3. Yamazaki, Kazuo, 2022. "Remarks on the non-uniqueness in law of the Navier–Stokes equations up to the J.-L. Lions’ exponent," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 226-269.
    4. David Criens, 2019. "Cylindrical Martingale Problems Associated with Lévy Generators," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1306-1359, September.

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