40. chen.lee.ea:26:strong
^^^^^^^^^^^^^^^^^^^^^^^^

.. centered:: Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations
.. centered:: Le Chen, Cheuk Yin Lee, and Panqiu Xia

**Abstract:** We study a class of stochastic time-fractional equations on
:math:`\mathbb{R}^d` driven by a centered Gaussian noise, involving a Caputo
time derivative of order :math:`\beta>0`, a fractional (power) Laplacian of
order :math:`\alpha>0`, and a Riemann-Liouville time integral of order
:math:`\gamma\ge0` acting on the noise. The noise is fractional in time (index
:math:`H`) and Riesz-type in space (index :math:`\ell`). We derive sharp
Dalang-type necessary and sufficient conditions for the existence of a random
field solution across almost full parameter range
:math:`(\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
:math:`\beta=1` and for parts of the case :math:`\beta=2`; one-sided version
for :math:`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 :math:`0` and :math:`\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.

`Preprint <https://arxiv.org/abs/2602.05317>`_

:cite:`chen.lee.ea:26:strong` 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*

.. code-block:: bibtex

   @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}
   }


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