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A note on intermittency for the fractional heat equation

Author

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  • Balan, Raluca M.
  • Conus, Daniel

Abstract

The goal of the present note is to study intermittency properties for the solution to the fractional heat equation ∂u∂t(t,x)=−(−Δ)β/2u(t,x)+u(t,x)Ẇ(t,x),t>0,x∈Rd with initial condition bounded above and below, where β∈(0,2] and the noise W behaves in time like a fractional Brownian motion of index H>1/2, and has a spatial covariance given by the Riesz kernel of index α∈(0,d). As a by-product, we obtain that the necessary and sufficient condition for the existence of the solution is α<β.

Suggested Citation

  • Balan, Raluca M. & Conus, Daniel, 2014. "A note on intermittency for the fractional heat equation," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 6-14.
    • Handle: RePEc:eee:stapro:v:95:y:2014:i:c:p:6-14
    DOI: 10.1016/j.spl.2014.08.001
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    Cited by:

    1. Yan, Litan & Yu, Xianye & Sun, Xichao, 2016. "Asymptotic behavior of the solution of the fractional heat equation," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 54-61.
    2. Song, Jian, 2017. "On a class of stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 37-79.
    3. Guerngar, Ngartelbaye & Nane, Erkan, 2020. "Moment bounds of a class of stochastic heat equations driven by space–time colored noise in bounded domains," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6246-6270.
    4. Litan Yan & Xianye Yu, 2019. "Asymptotic Behavior for High Moments of the Fractional Heat Equation with Fractional Noise," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1617-1646, December.

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