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On Some Properties of a Class of Fractional Stochastic Heat Equations

Author

Listed:
  • Wei Liu

    (Shanghai Normal University)

  • Kuanhou Tian

    (Loughborough University)

  • Mohammud Foondun

    (Loughborough University)

Abstract

We consider nonlinear parabolic stochastic equations of the form $$\partial _t u=\mathcal {L}u + \lambda \sigma (u)\dot{\xi }$$ ∂ t u = L u + λ σ ( u ) ξ ˙ on the ball $$B(0,\,R)$$ B ( 0 , R ) , where $$\dot{\xi }$$ ξ ˙ denotes some Gaussian noise and $$\sigma $$ σ is Lipschitz continuous. Here $$\mathcal {L}$$ L corresponds to a symmetric $$\alpha $$ α -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on $$\sigma $$ σ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).

Suggested Citation

  • Wei Liu & Kuanhou Tian & Mohammud Foondun, 2017. "On Some Properties of a Class of Fractional Stochastic Heat Equations," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1310-1333, December.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0684-6
    DOI: 10.1007/s10959-016-0684-6
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Kumar, Vivek, 2022. "Stochastic fractional heat equation perturbed by general Gaussian and non-Gaussian noise," Statistics & Probability Letters, Elsevier, vol. 184(C).

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