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The heat equation with Lévy noise

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  • Mueller, Carl

Abstract

We prove short-time existence for parabolic equations with Lévy noise of the form where is nonnegative Lévy noise of index is the power of the Laplacian, , and is a continuous nonnegative function. is a bounded open domain in . A sufficient condition for short time existence is While we cannot prove uniqueness, we show that the solution we construct is minimal among all solutions.

Suggested Citation

  • Mueller, Carl, 1998. "The heat equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 67-82, May.
    • Handle: RePEc:eee:spapps:v:74:y:1998:i:1:p:67-82
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    Citations

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    Cited by:

    1. Chong, Carsten, 2017. "Stochastic PDEs with heavy-tailed noise," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2262-2280.
    2. Debbi, Latifa & Dozzi, Marco, 2005. "On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1764-1781, November.
    3. Funaki, Tadahisa & Xie, Bin, 2009. "A stochastic heat equation with the distributions of Lévy processes as its invariant measures," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 307-326, February.
    4. Jacob, Niels & Potrykus, Alexander & Wu, Jiang-Lun, 2010. "Solving a non-linear stochastic pseudo-differential equation of Burgers type," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2447-2467, December.
    5. Mueller, Carl & Mytnik, Leonid & Stan, Aurel, 2006. "The heat equation with time-independent multiplicative stable Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 70-100, January.
    6. 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.
    7. Liang, Fei & Zhang, Yidi & Zhao, Shuangshuang, 2024. "Blow-up solution to an abstract non-local stochastic heat equation with Lévy noise," Statistics & Probability Letters, Elsevier, vol. 206(C).
    8. Anh, V. V. & Leonenko, N. N., 1999. "Non-Gaussian scenarios for the heat equation with singular initial conditions," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 91-114, November.
    9. Carsten Chong, 2017. "Lévy-driven Volterra Equations in Space and Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1014-1058, September.

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