Mild Solutions for Time--Fractional Stochastic Nonlocal Diffusion Equations

TL;DR

This work addresses the well-posedness of mild solutions for a time-fractional stochastic diffusion equation with a nonlocal spatial operator and additive time--space white noise. By applying Laplace and Fourier transforms, the authors obtain an explicit mild-solution representation $\mathbf{Z}(t,x)=\mathbf{Z}_1(t,x)+\mathbf{Z}_2(t,x)$, where $\mathbf{Z}_1$ involves $E_{\alpha}(-a(\xi)t^{\alpha})$ and $\mathbf{Z}_2$ is a stochastic convolution with the kernel $\Lambda(t-\tau,\xi)$, and they prove a sharp classification of mild solutions in terms of $(\alpha,N,\lambda,\mu)$. The key contributions are the complete mildness criteria, the nonexistence result in the purely nonlocal case, and the regime-dependent existence in the local+nonlocal setting, complemented by numerical simulations that demonstrate subdiffusive spreading and memory effects on the mean and variance. These results extend known findings for purely local fractional stochastic heat equations by incorporating nonlocal spatial diffusion and provide rigorous thresholds for well-posedness in physically relevant regimes. The findings have potential implications for modeling anomalous transport with memory in spatially extended systems.

Abstract

We study a time--space nonlocal diffusion equation driven by additive time--space white noise, where the time derivative is the Caputo derivative of order $α\in(0,2)$. The model couples local diffusion with a nonlocal convolution operator generated by a radial probability density, thus incorporating memory effects and long-range spatial interactions. For Dirac initial data, we derive an explicit solution formula in the space of tempered distributions, decomposing the solution into a deterministic part and a stochastic convolution kernel expressed through Mittag--Leffler functions. Our main contribution is a sharp characterization of the existence of mild solutions in terms of $α$, the spatial dimension $N$, and the coefficients of the local and nonlocal diffusion terms. In particular, when the Laplacian term is absent, no mild solution exists, whereas for $λ>0$ the admissible regimes depend critically on $(α,N)$, extending and sharpening the known results for purely local fractional stochastic heat equations. Numerical simulations illustrate the evolution of the mean and variance and emphasize subdiffusive spreading and memory effects.

Figures (5)

• Figure 1: Mean solution $\mathbf{Z}_0(t,x)$ for the classical case $\alpha=1$.
• Figure 2: Variance $\operatorname{Var}[\mathbf{Z}(t,x)]$ for the classical case $\alpha=1$.
• Figure 3: Mean solution $\mathbf{Z}_0(t,x)$, $\alpha=0.6, \lambda=1$.
• Figure 4: Decay of the coefficients $\beta_m$ versus $m$.
• Figure 5: Variance $\operatorname{Var}[\mathbf{Z}(t,x)]$ for $\alpha=0.6$ and $\lambda=1$ (e.g. $\sigma=1$).

Theorems & Definitions (16)

• Theorem 1.1
• Definition 1.1
• Theorem 1.2
• Remark 1.1
• Definition 2.1
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Proposition 2.1
• Theorem 2.2