Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations

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.

💡 Research Summary

The paper investigates a broad class of stochastic partial differential equations (SPDEs) on ℝⁿ that incorporate three fractional operators: a Caputo time‑derivative of order β > 0, a fractional Laplacian (−Δ)^{α/2} of order α > 0, and a Riemann–Liouville time integral of order γ ≥ 0 acting on the driving noise. The noise itself is a centered Gaussian field that is fractional in time with Hurst index H and Riesz‑type in space with index ℓ. The main contributions can be grouped into four pillars: (i) existence and Dalang‑type integrability condition, (ii) sharp asymptotics of the fundamental solution kernels, (iii) variance bounds and strong local nondeterminism (SLND) in both time and space, and (iv) applications to fine sample‑path properties such as uniform and local moduli of continuity, Chung‑type laws of the iterated logarithm, and small‑ball probability estimates.

Existence and Dalang‑type condition. By representing the mild solution as a stochastic convolution
u(t,x)=∫₀ᵗ∫{ℝⁿ} G{β,α,γ}(t−s,x−y) Ẇ(ds,dy),
the authors reduce the problem to the square‑integrability of the kernel in the spectral domain. Using Fourier analysis they obtain the necessary and sufficient condition
∫_{ℝⁿ} |ξ|^{-2ℓ} (1+|ξ|^{α})^{-2β} dξ < ∞,
which generalises the classical Dalang condition for the stochastic heat and wave equations. The condition is shown to be essentially optimal across the whole parameter range (α,β,γ;H,ℓ).

Fundamental solution asymptotics. The kernel G_{β,α,γ} is expressed in terms of Fox H‑functions and Mittag‑Leffler functions. The authors derive precise two‑sided estimates for G near the origin (t→0, |x|→0) and at infinity (t→∞ or |x|→∞). Near the origin,
G(t,x) ≍ t^{β−1−γ} |x|^{α−d},
while at large arguments the kernel decays like a stretched exponential combined with a polynomial prefactor, reflecting the interplay between the fractional diffusion and the memory term. These asymptotics are crucial for the subsequent variance calculations.

Variance bounds and SLND. Using the kernel estimates, the paper obtains sharp two‑sided bounds for temporal and spatial increments:
E|u(t+h,x)−u(t,x)|² ≍ h^{2βH+2γ−1} (1+|x|)^{-2ℓ},
E|u(t,x+z)−u(t,x)|² ≍ |z|^{2αH+2γ−d} (1+t)^{-2βH}.
From these bounds the authors prove strong local nondeterminism. In time, a two‑sided SLND holds for β=1 (the stochastic heat equation) and for parts of the β=2 case; a one‑sided version holds for all 0<β<2. In space, SLND holds uniformly for the entire parameter range. The SLND constants are independent of the number of conditioning points, which is essential for fine path analysis.

Sample‑path applications. Leveraging SLND, the authors derive:

Uniform modulus of continuity:
sup_{|t−s|≤δ,|x−y|≤δ}|u(t,x)−u(s,y)| ≤ C δ^{βH+γ−½} (log 1/δ)^{½} a.s.

Local modulus (Chung‑type LIL):
limsup_{δ→0} sup_{|t−s|≤δ}|u(t,x)−u(s,x)| / (δ^{βH+γ−½}√{2 log log 1/δ}) = 1 a.s.

Small‑ball probabilities:
P( sup_{t∈