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Asymptotic Behavior for High Moments of the Fractional Heat Equation with Fractional Noise

Author

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  • Litan Yan

    (Donghua University)

  • Xianye Yu

    (Zhejiang Gongshang University)

Abstract

In this paper, we investigate the large time behavior of the solution to the fractional heat equation $$\begin{aligned} \frac{\partial u}{\partial t}(t,x)=-(-\Delta )^{\beta /2}u(t,x)+u(t,x)\frac{\partial ^{d+1} W}{\partial t\partial x_1\cdots \partial x_d},\quad t>0,\quad x\in \mathbb {R}^d, \end{aligned}$$ ∂ u ∂ t ( t , x ) = - ( - Δ ) β / 2 u ( t , x ) + u ( t , x ) ∂ d + 1 W ∂ t ∂ x 1 ⋯ ∂ x d , t > 0 , x ∈ R d , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) and the noise W(t, x) is a fractional Brownian sheet with indexes $$H_0, H_1,\ldots ,H_d\in (\frac{1}{2},1)$$ H 0 , H 1 , … , H d ∈ ( 1 2 , 1 ) . By using large deviation techniques and variational method, we find a constant $$M_1$$ M 1 such that for any integer $$p\ge 1$$ p ≥ 1 and $$\alpha _0\beta +\alpha

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:4:d:10.1007_s10959-019-00899-9
    DOI: 10.1007/s10959-019-00899-9
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    References listed on IDEAS

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    1. Song, Jian, 2012. "Asymptotic behavior of the solution of heat equation driven by fractional white noise," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 614-620.
    2. Chen, Xia & Rosen, Jay, 2010. "Large deviations and renormalization for Riesz potentials of stable intersection measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1837-1878, August.
    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. 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.
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