Mild Solutions for Time--Fractional Stochastic Nonlocal Diffusion Equations
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909996801851392 |
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| author | Alwohaibi, M. Alsaleh, D. El-Beltagy, M. Majdoub, M. Mliki, E. |
| author_facet | Alwohaibi, M. Alsaleh, D. El-Beltagy, M. Majdoub, M. Mliki, E. |
| contents | We study a time--space nonlocal diffusion equation driven by additive time--space white noise, where the time derivative is the Caputo derivative of order $α\in(0,2)$. The model couples local diffusion with a nonlocal convolution operator generated by a radial probability density, thus incorporating memory effects and long-range spatial interactions.
For Dirac initial data, we derive an explicit solution formula in the space of tempered distributions, decomposing the solution into a deterministic part and a stochastic convolution kernel expressed through Mittag--Leffler functions. Our main contribution is a sharp characterization of the existence of mild solutions in terms of $α$, the spatial dimension $N$, and the coefficients of the local and nonlocal diffusion terms. In particular, when the Laplacian term is absent, no mild solution exists, whereas for $λ>0$ the admissible regimes depend critically on $(α,N)$, extending and sharpening the known results for purely local fractional stochastic heat equations. Numerical simulations illustrate the evolution of the mean and variance and emphasize subdiffusive spreading and memory effects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14838 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mild Solutions for Time--Fractional Stochastic Nonlocal Diffusion Equations Alwohaibi, M. Alsaleh, D. El-Beltagy, M. Majdoub, M. Mliki, E. Analysis of PDEs Probability We study a time--space nonlocal diffusion equation driven by additive time--space white noise, where the time derivative is the Caputo derivative of order $α\in(0,2)$. The model couples local diffusion with a nonlocal convolution operator generated by a radial probability density, thus incorporating memory effects and long-range spatial interactions. For Dirac initial data, we derive an explicit solution formula in the space of tempered distributions, decomposing the solution into a deterministic part and a stochastic convolution kernel expressed through Mittag--Leffler functions. Our main contribution is a sharp characterization of the existence of mild solutions in terms of $α$, the spatial dimension $N$, and the coefficients of the local and nonlocal diffusion terms. In particular, when the Laplacian term is absent, no mild solution exists, whereas for $λ>0$ the admissible regimes depend critically on $(α,N)$, extending and sharpening the known results for purely local fractional stochastic heat equations. Numerical simulations illustrate the evolution of the mean and variance and emphasize subdiffusive spreading and memory effects. |
| title | Mild Solutions for Time--Fractional Stochastic Nonlocal Diffusion Equations |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2601.14838 |