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Stochastic PDEs with heavy-tailed noise

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  • Chong, Carsten

Abstract

We analyze the nonlinear stochastic heat equation driven by heavy-tailed noise on unbounded domains and in arbitrary dimension. The existence of a solution is proved even if the noise only has moments up to an order strictly smaller than its Blumenthal–Getoor index. In particular, this includes all stable noises with index α<1+2/d. Although we cannot show uniqueness, the constructed solution is natural in the sense that it is the limit of the solutions to approximative equations obtained by truncating the big jumps of the noise or by restricting its support to a compact set in space. Under growth conditions on the nonlinear term we can further derive moment estimates of the solution, uniformly in space. Finally, the techniques are shown to apply to Volterra equations with kernels bounded by generalized Gaussian densities. This includes, for instance, a large class of uniformly parabolic stochastic PDEs.

Suggested Citation

  • Chong, Carsten, 2017. "Stochastic PDEs with heavy-tailed noise," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2262-2280.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:7:p:2262-2280
    DOI: 10.1016/j.spa.2016.10.011
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    References listed on IDEAS

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    1. Mueller, Carl, 1998. "The heat equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 67-82, May.
    2. Albeverio, Sergio & Wu, Jiang-Lun & Zhang, Tu-Sheng, 1998. "Parabolic SPDEs driven by Poisson white noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 21-36, May.
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    Cited by:

    1. Tomasz Kosmala & Markus Riedle, 2021. "Stochastic Integration with Respect to Cylindrical Lévy Processes by p-Summing Operators," Journal of Theoretical Probability, Springer, vol. 34(1), pages 477-497, March.
    2. Pham, Viet Son & Chong, Carsten, 2018. "Volterra-type Ornstein–Uhlenbeck processes in space and time," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 3082-3117.
    3. Harin, Alexander, 2018. "Forbidden zones for the expectation. New mathematical results for behavioral and social sciences," MPRA Paper 86650, University Library of Munich, Germany.
    4. Kosmala, Tomasz & Riedle, Markus, 2022. "Stochastic evolution equations driven by cylindrical stable noise," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 278-307.

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