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Some effects of the noise intensity upon non-linear stochastic heat equations on [0,1]

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  • Xie, Bin

Abstract

Various effects of the noise intensity upon the solution u(t,x) of the stochastic heat equation with Dirichlet boundary conditions on [0,1] are investigated. We show that for small noise intensity, the pth moment of supx∈[0,1]|u(t,x)| is exponentially stable, however, for large one, it grows at least exponentially. We also prove that the noise excitation of the pth energy of u(t,x) is 4, as the noise intensity goes to infinity. We formulate a common method to investigate the lower bounds of the above two different behaviors for large noise intensity, which are hard parts in Foondun and Joseph (2014), Foondun and Nualart (2015) and Khoshnevisan and Kim (2015).

Suggested Citation

  • Xie, Bin, 2016. "Some effects of the noise intensity upon non-linear stochastic heat equations on [0,1]," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1184-1205.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:4:p:1184-1205
    DOI: 10.1016/j.spa.2015.10.014
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    References listed on IDEAS

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    1. Zhang, Tusheng, 2012. "Large deviations for invariant measures of SPDEs with two reflecting walls," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3425-3444.
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    Cited by:

    1. Harin, Alexander, 2018. "Forbidden zones for the expectation. New mathematical results for behavioral and social sciences," MPRA Paper 86650, University Library of Munich, Germany.
    2. 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.
    3. Foondun, Mohammud & Guerngar, Ngartelbaye & Nane, Erkan, 2017. "Some properties of non-linear fractional stochastic heat equations on bounded domains," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 86-93.

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