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Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

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  • Röckner, Michael
  • Zhu, Rongchan
  • Zhu, Xiangchan

Abstract

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.

Suggested Citation

  • Röckner, Michael & Zhu, Rongchan & Zhu, Xiangchan, 2014. "Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1974-2002.
  • Handle: RePEc:eee:spapps:v:124:y:2014:i:5:p:1974-2002
    DOI: 10.1016/j.spa.2014.01.010
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    References listed on IDEAS

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    1. Liu, Wei & Röckner, Michael & Zhu, Xiang-Chan, 2013. "Large deviation principles for the stochastic quasi-geostrophic equations," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3299-3327.
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    Cited by:

    1. Lü, Huaxiang & Zhu, Xiangchan, 2023. "Global-in-time probabilistically strong solutions to stochastic power-law equations: Existence and non-uniqueness," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 62-98.
    2. Li, Jingna & Liu, Hongxia & Tang, Hao, 2021. "Stochastic MHD equations with fractional kinematic dissipation and partial magnetic diffusion in R2," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 139-182.
    3. Xu, Liping & Li, Zhi, 2018. "Stochastic fractional evolution equations with fractional brownian motion and infinite delay," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 36-46.
    4. Yanmei Liu & Monzorul Khan & Yubin Yan, 2016. "Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations," Mathematics, MDPI, vol. 4(3), pages 1-28, July.

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