Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity
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| Format: | Preprint |
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2023
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| _version_ | 1866911305436233728 |
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| author | Liu, Shuhui Hu, Yaozhong Wang, Xiong |
| author_facet | Liu, Shuhui Hu, Yaozhong Wang, Xiong |
| contents | We determine the range of Hurst parameters that provide the necessary and sufficient conditions for the solvability, in $L^2(Ω)$, of the stochastic wave equation: $ \frac{\partial^2 }{\partial t^2}u(t,x) =Δu(t,x)+\dot{W}(t,x)$, where $\{ W(t,x),\ t\geq 0, x\in \mathbb{R}^d\} $ is a fractional Brownian field with temporal Hurst parameter $H_0\in[\tfrac12,1]$ and spatial Hurst parameters $H_i\in(0,1)$ for $i=1,\cdots,d$. {In particular, the solvability condition exhibits a phase transition at $H_0 = 1$.} We also obtain the sharp growth rate and the sharp Hölder continuity of the solution on the real line in the case $H_0=1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_02425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity Liu, Shuhui Hu, Yaozhong Wang, Xiong Probability Analysis of PDEs Primary 60H15, secondary 60H05, 60G15, 60G22 We determine the range of Hurst parameters that provide the necessary and sufficient conditions for the solvability, in $L^2(Ω)$, of the stochastic wave equation: $ \frac{\partial^2 }{\partial t^2}u(t,x) =Δu(t,x)+\dot{W}(t,x)$, where $\{ W(t,x),\ t\geq 0, x\in \mathbb{R}^d\} $ is a fractional Brownian field with temporal Hurst parameter $H_0\in[\tfrac12,1]$ and spatial Hurst parameters $H_i\in(0,1)$ for $i=1,\cdots,d$. {In particular, the solvability condition exhibits a phase transition at $H_0 = 1$.} We also obtain the sharp growth rate and the sharp Hölder continuity of the solution on the real line in the case $H_0=1/2$. |
| title | Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity |
| topic | Probability Analysis of PDEs Primary 60H15, secondary 60H05, 60G15, 60G22 |
| url | https://arxiv.org/abs/2305.02425 |