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Moment bounds of a class of stochastic heat equations driven by space–time colored noise in bounded domains

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  • Guerngar, Ngartelbaye
  • Nane, Erkan

Abstract

We consider the fractional stochastic heat type equation ∂∂tut(x)=−(−Δ)α∕2ut(x)+ξσ(ut(x))Ḟ(t,x),x∈D,t>0,with nonnegative bounded initial condition, where α∈(0,2], ξ>0 is the noise level, σ:R→R is a globally Lipschitz function satisfying some growth conditions and the noise term Ḟ behaves in space like the Riez kernel and is possibly correlated in time and D is the unit open ball centered at the origin in Rd. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level ξ. On the other hand when the noise term behaves in time like the fractional Brownian motion with index H∈(1∕2,1), We also derive explicit bounds leading to a well-known intermittency property.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:6246-6270
    DOI: 10.1016/j.spa.2020.05.009
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    References listed on IDEAS

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    1. 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.
    2. Balan, Raluca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2017. "Intermittency for the Hyperbolic Anderson Model with rough noise in space," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2316-2338.
    3. 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.
    4. 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.
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