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Solving a non-linear stochastic pseudo-differential equation of Burgers type

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  • Jacob, Niels
  • Potrykus, Alexander
  • Wu, Jiang-Lun

Abstract

In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form [not partial differential]tu+q(x,D)u+[not partial differential]xf(t,x,u)=h1(t,x,u)+h2(t,x,u)Ft,x for , where q(x,D) is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, are measurable functions, and Ft,x stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space in the mild formulation and show the existence of a unique local mild solution to the initial value problem by utilising a fixed point argument.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:12:p:2447-2467
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    References listed on IDEAS

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    1. Chen, Zhen-Qing & Kumagai, Takashi, 2003. "Heat kernel estimates for stable-like processes on d-sets," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 27-62, November.
    2. Mueller, Carl, 1998. "The heat equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 67-82, May.
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    Cited by:

    1. Junfeng Liu & Litan Yan, 2016. "Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise," Journal of Theoretical Probability, Springer, vol. 29(1), pages 307-347, March.

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