Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations
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| Format: | Preprint |
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2026
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| author | Chen, Le Lee, Cheuk Yin Xia, Panqiu |
| author_facet | Chen, Le Lee, Cheuk Yin Xia, Panqiu |
| contents | We study a class of stochastic time-fractional equations on $\mathbb{R}^d$ driven by a centered Gaussian noise, involving a Caputo time derivative of order $β>0$, a fractional (power) Laplacian of order $α>0$, and a Riemann-Liouville time integral of order $γ\ge0$ acting on the noise. The noise is fractional in time (index $H$) and Riesz-type in space (index $\ell$). We derive sharp Dalang-type necessary and sufficient conditions for the existence of a random field solution across almost full parameter range $(α,β,γ;H,\ell)$. Under the Dalang-type conditions, we prove sharp variance bounds for temporal and spatial increments, as well as strong local nondeterminism in time in several regimes (two-sided version for $β=1$ and for parts of the case $β=2$; one-sided version for $0<β<2$) and strong local nondeterminism in space for the whole range of parameters. As applications, we derive exact uniform and local moduli of continuity, Chung-type laws of the iterated logarithm, and quantitative bounds on small ball probabilities. Along the way, we obtain sharp asymptotics for the fundamental solution kernels at $0$ and $\infty$, which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_05317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations Chen, Le Lee, Cheuk Yin Xia, Panqiu Probability Mathematical Physics Analysis of PDEs 60G17, 60H15 (primary), 60G15, 60G22, 33G12 (secondary) We study a class of stochastic time-fractional equations on $\mathbb{R}^d$ driven by a centered Gaussian noise, involving a Caputo time derivative of order $β>0$, a fractional (power) Laplacian of order $α>0$, and a Riemann-Liouville time integral of order $γ\ge0$ acting on the noise. The noise is fractional in time (index $H$) and Riesz-type in space (index $\ell$). We derive sharp Dalang-type necessary and sufficient conditions for the existence of a random field solution across almost full parameter range $(α,β,γ;H,\ell)$. Under the Dalang-type conditions, we prove sharp variance bounds for temporal and spatial increments, as well as strong local nondeterminism in time in several regimes (two-sided version for $β=1$ and for parts of the case $β=2$; one-sided version for $0<β<2$) and strong local nondeterminism in space for the whole range of parameters. As applications, we derive exact uniform and local moduli of continuity, Chung-type laws of the iterated logarithm, and quantitative bounds on small ball probabilities. Along the way, we obtain sharp asymptotics for the fundamental solution kernels at $0$ and $\infty$, which may be of independent interest. |
| title | Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations |
| topic | Probability Mathematical Physics Analysis of PDEs 60G17, 60H15 (primary), 60G15, 60G22, 33G12 (secondary) |
| url | https://arxiv.org/abs/2602.05317 |