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The stochastic wave equation with fractional noise: A random field approach

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  • Balan, Raluca M.
  • Tudor, Ciprian A.

Abstract

We consider the linear stochastic wave equation with spatially homogeneous Gaussian noise, which is fractional in time with index H>1/2. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in Dalang (1999) [10], where the noise is white in time. Under this condition, we show that the solution is L2([Omega])-continuous. Similar results are obtained for the heat equation. Unlike in the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.

Suggested Citation

  • Balan, Raluca M. & Tudor, Ciprian A., 2010. "The stochastic wave equation with fractional noise: A random field approach," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2468-2494, December.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:12:p:2468-2494
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    References listed on IDEAS

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    1. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    2. Nualart, Eulalia & Viens, Frederi, 2009. "The fractional stochastic heat equation on the circle: Time regularity and potential theory," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1505-1540, May.
    3. Quer-Sardanyons, Lluís & Tindel, Samy, 2007. "The 1-d stochastic wave equation driven by a fractional Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1448-1472, October.
    4. Dalang, Robert C. & Hou, Qiang, 1997. "On Markov properties of Lévy waves in two dimensions," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 265-287, December.
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    Cited by:

    1. Zhang, Yinghan & Yang, Xiaoyuan, 2015. "Fractional stochastic Volterra equation perturbed by fractional Brownian motion," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 20-36.

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