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

Litan Yan () and Xianye Yu ()
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Litan Yan: Donghua University
Xianye Yu: Zhejiang Gongshang University

Journal of Theoretical Probability, 2019, vol. 32, issue 4, 1617-1646

Abstract: 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

Keywords: Fractional heat equation; Fractional Brownian sheet; Asymptotic behavior; 60H15; 35B40 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-019-00899-9

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