Space-time fractional stochastic partial differential equations driven by Lévy white noise
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911007111118848 |
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| author | Guo, Yuhui Wu, Jiang-Lun |
| author_facet | Guo, Yuhui Wu, Jiang-Lun |
| contents | This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation \begin{equation*}
\left(\partial_t^β+\fracν{2}\left(-Δ\right)^{α/ 2}\right) u=I_{t}^γ\Big[ f(t,x,u)-\sum_{i=1}^{d} \frac{\partial}{\partial x_i} q_i(t,x,u)+ σ(t,x,u) F_{t,x}\Big] \end{equation*} for a random field $u(t,x):[0,\infty)\times\mathbb{R}^d \mapsto\mathbb{R}$, where $α>0, β\in(0,2), γ\ge0, ν>0, F_{t,x}$ is a Lévy space-time white noise, $I_{t}^γ$ stands for the Riemann-Liouville integral in time, and $f,q_i,σ:[0,\infty)\times\mathbb{R}^d\times\mathbb{R} \mapsto\mathbb{R}$ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of $L^2(\mathbb{R}^d)$-valued local solutions when the Lévy white noise $F_{t,x}$ contains Gaussian noise component. Furthermore, for $p\in[1,2]$, we derive the existence and uniqueness of $L^p(\mathbb{R}^d)$-valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12834 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Space-time fractional stochastic partial differential equations driven by Lévy white noise Guo, Yuhui Wu, Jiang-Lun Probability Primary 60H15, Secondary 60G51, 26A33 This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation \begin{equation*} \left(\partial_t^β+\fracν{2}\left(-Δ\right)^{α/ 2}\right) u=I_{t}^γ\Big[ f(t,x,u)-\sum_{i=1}^{d} \frac{\partial}{\partial x_i} q_i(t,x,u)+ σ(t,x,u) F_{t,x}\Big] \end{equation*} for a random field $u(t,x):[0,\infty)\times\mathbb{R}^d \mapsto\mathbb{R}$, where $α>0, β\in(0,2), γ\ge0, ν>0, F_{t,x}$ is a Lévy space-time white noise, $I_{t}^γ$ stands for the Riemann-Liouville integral in time, and $f,q_i,σ:[0,\infty)\times\mathbb{R}^d\times\mathbb{R} \mapsto\mathbb{R}$ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of $L^2(\mathbb{R}^d)$-valued local solutions when the Lévy white noise $F_{t,x}$ contains Gaussian noise component. Furthermore, for $p\in[1,2]$, we derive the existence and uniqueness of $L^p(\mathbb{R}^d)$-valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions. |
| title | Space-time fractional stochastic partial differential equations driven by Lévy white noise |
| topic | Probability Primary 60H15, Secondary 60G51, 26A33 |
| url | https://arxiv.org/abs/2506.12834 |