The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914262550577152 |
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| author | Yang, Yong Zhen Zhou, Yong |
| author_facet | Yang, Yong Zhen Zhou, Yong |
| contents | This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--Lévy noises. The equations feature a Caputo time derivative $\partial_t^α$ ($0<α<1$) and a spatial nonlocal operator $ϕ(Δ)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure.
For the case $p \ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $ϕ$-Sobolev spaces $\mathcal{H}_p^{ϕ,γ+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and Lévy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d < 2κ_0\bigl(2 - (2σ_2 - 2/p)_+/α\bigr)$ is obtained.~For the low-regularity case $1 \le p \le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p(\mathbb{R}^d)$ for equations driven by pure-jump Lévy space-time white noise, using stochastic truncation and fixed-point arguments.
The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_03754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$ Yang, Yong Zhen Zhou, Yong Analysis of PDEs Probability 26A33, 35R11 This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--Lévy noises. The equations feature a Caputo time derivative $\partial_t^α$ ($0<α<1$) and a spatial nonlocal operator $ϕ(Δ)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. For the case $p \ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $ϕ$-Sobolev spaces $\mathcal{H}_p^{ϕ,γ+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and Lévy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d < 2κ_0\bigl(2 - (2σ_2 - 2/p)_+/α\bigr)$ is obtained.~For the low-regularity case $1 \le p \le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p(\mathbb{R}^d)$ for equations driven by pure-jump Lévy space-time white noise, using stochastic truncation and fixed-point arguments. The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness. |
| title | The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$ |
| topic | Analysis of PDEs Probability 26A33, 35R11 |
| url | https://arxiv.org/abs/2512.03754 |