40. chen.lee.ea:26:strong#
Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations
Le Chen, Cheuk Yin Lee, and Panqiu Xia
Abstract: 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 \(\beta>0\), a fractional (power) Laplacian of order \(\alpha>0\), and a Riemann-Liouville time integral of order \(\gamma\ge0\) 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 \((\alpha,\beta,\gamma;H,\ell)\). 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 \(\beta=1\) and for parts of the case \(\beta=2\); one-sided version for \(0<\beta<2\)) 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.
MSC 2020 subject classifications: Primary 60G17, 60H15; Secondary 60G15, 60G22, 33G12.
Keywords: strong local nondeterminism, stochastic time-fractional diffusion equations, Caputo derivative, fractional Laplacian, Dalang condition, moduli of continuity, Chung’s law of the iterated logarithm, small ball probabilities.
[CLX26] Le Chen, Cheuk Yin Lee & Panqiu Xia (2026) ‘Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations’, preprint arXiv:2602.05317, 125 pages, 6 figures, 1 table
@article{chen.lee.ea:26:strong,
title = {Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations},
author = {Le Chen and Cheuk Yin Lee and Panqiu Xia},
year = {2026},
month = {February},
journal = {Preprint arXiv:2602.05317},
url = {http://arXiv.org/abs/2602.05317}
}