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

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 $β>0$, 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.

Figures (6)

• Figure 1: Three cases and some special cases in Theorem \ref{['T:Dalang']}.
• Figure 2: (SHE with white noise in time) Plots of the constant in $K_{\ref{['E:K+1']}}$ as a function of $\ell$ for $\alpha = 3/2$, $2$, and $3$, with special values when $\ell = \alpha/2$. Note that $\ell = \alpha/2$ are not the minimums of the respective curves.
• Figure 3: (SWE with white noise in time) Plots of the constant in $K_{\ref{['E:K-1']}}$ as a function of $\ell$ for $\alpha = 3/2$, $2$, and $3$.
• Figure 4: (SWE with critical spatial noise) Plots of the constants in $K_{\ref{['E:K-5']}}$ as functions of $H$.
• Figure 5: Plot of function $\widehat{h}(\xi)$.

Theorems & Definitions (101)

• Theorem 1.1: Sharp well-posedness
• proof
• Theorem 1.2: Two-sided variance bounds
• proof
• Remark 1.1
• Theorem 1.3: Strong local nondeterminism (SLND)
• proof
• Conjecture 1.1
• Remark 1.2
• Corollary 1.1: Uniform moduli of continuity