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Mild Solutions for Time--Fractional Stochastic Nonlocal Diffusion Equations

This paper establishes the existence and explicit representation of mild solutions for time-space nonlocal diffusion equations driven by additive white noise with Caputo time derivatives, providing a sharp characterization of solvability regimes based on the fractional order, spatial dimension, and the interplay between local and nonlocal diffusion terms.

Original authors: M. Alwohaibi, D. Alsaleh, M. El-Beltagy, M. Majdoub, E. Mliki

Published 2026-01-22

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CC BY 4.0

Original authors: M. Alwohaibi, D. Alsaleh, M. El-Beltagy, M. Majdoub, E. Mliki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a drop of ink spread out in a glass of water. In the "classical" world (the way physics usually works), the ink spreads out smoothly and predictably, like a perfect circle getting bigger every second. This is called diffusion.

Now, imagine two things change:

The water is "sticky" or "forgetful": The ink doesn't just move based on where it is right now; it remembers where it was a moment ago. This is fractional time.
The water is "jumpy": Instead of just flowing, the ink particles sometimes get kicked by invisible, random forces (like tiny, invisible billiard balls hitting them). This is stochastic noise.

This paper studies a very complex version of that ink-drop scenario. The authors are trying to figure out: "Under what specific conditions does this messy, sticky, jumpy ink drop actually have a mathematically valid solution?"

Here is the breakdown of their findings using simple analogies:

1. The Setup: A "Double-Action" Diffusion

The equation they study describes a particle that moves in two ways at once:

Local Diffusion (The "Smooth Slide"): Like the ink sliding through water normally. This is controlled by a term called the Laplacian.
Nonlocal Diffusion (The "Teleportation"): The particle can also "jump" to a faraway spot instantly, but only if there is a specific probability map (a "radial density") telling it where it's likely to land. This is the nonlocal part.

They add random noise (the "jumpy" part) and memory (the "sticky" part) to this mix.

2. The Big Question: Does a Solution Exist?

In math, just writing down an equation doesn't mean it has a real answer. Sometimes, the math explodes into infinity, meaning the physical situation described is impossible to define.

The authors asked: "When can we actually calculate the position of this ink drop?"

They found that the answer depends entirely on the balance between the "Smooth Slide" (Local) and the "Teleportation" (Nonlocal), and how "sticky" the time is (the fractional order $\alpha$ ).

3. The Surprising Discovery: The "Smooth Slide" is Essential

The most striking result is this: If you remove the "Smooth Slide" (the local diffusion) and rely only on the "Teleportation" (nonlocal) plus the random noise, the math breaks completely.

Analogy: Imagine trying to walk across a room where you can only teleport to random spots, but you are also being pushed by a chaotic wind. If you have no ability to walk normally (no local diffusion), you will never settle into a predictable pattern. The "ink" becomes so chaotic that it cannot be measured or defined.
The Paper's Claim: If the local diffusion term is zero, no solution exists, no matter how many dimensions you are in or how "sticky" the time is.

4. The "Goldilocks" Zone: When the "Smooth Slide" is Present

If you do have the "Smooth Slide" (local diffusion), then a solution can exist, but only in very specific "Goldilocks" zones depending on the dimension (how many directions the ink can move) and the memory (how fractional the time is).

The "Memory" Factor ( $\alpha$ ):
- If the memory is very strong (time is very fractional, $\alpha$ is small), the ink spreads very slowly (subdiffusion).
- If the memory is weak (closer to normal time, $\alpha$ is large), it spreads faster.
The Dimension Rule:
- In 1D (a line): You can have a solution if the memory is strong enough (specifically, if the "stickiness" is between a certain range).
- In 2D (a flat surface): You can have a solution if the time is "super-memory" (between 1 and 2).
- In 3D or higher: The math says the ink becomes too chaotic to define a solution in these specific nonlocal setups.

5. What the Simulations Showed

The authors ran computer simulations to visualize this:

The "Normal" Case: The ink spreads out like a bell curve (Gaussian), getting wider and flatter over time.
The "Fractional" Case: The ink spreads much slower. It stays clumped near the center for a long time before slowly leaking out. This is called subdiffusion.
The "Variance" (Chaos): They measured how "jumpy" the ink was. In the normal world, the chaos grows steadily. In the fractional world, the chaos grows, but the "memory" of the system makes the spreading process much more sluggish and "heavy-tailed" (meaning the ink stays closer to the center longer than expected).

Summary

Think of this paper as a rulebook for a very strange game of "Ink Drop."

Rule 1: You must have a "normal walking" ability (local diffusion). If you only have "teleportation" and "random wind," the game is broken; no solution exists.
Rule 2: If you have "normal walking," you can play, but only if the "stickiness" of time and the size of the room (dimensions) match up perfectly.
Result: When the rules are followed, the ink spreads slowly and remembers its past, creating a unique, "sub-diffusive" pattern that is very different from the smooth spreading we see in everyday life.

The paper provides the exact mathematical formulas to predict when this game works and when it fails, using special functions (Mittag-Leffler) that act like the "exponential function" for this sticky, fractional world.

Technical Summary: Mild Solutions for Time–Fractional Stochastic Nonlocal Diffusion Equations

Problem Statement
The paper investigates a time–space nonlocal diffusion equation driven by additive time–space white noise. The model is governed by the Caputo fractional derivative of order $\alpha \in (0, 2)$ and couples a local diffusion term (Laplacian) with a nonlocal convolution operator generated by a radial probability density. The governing equation is:
$\partial_t^\alpha Z(t, x) = \lambda \Delta Z(t, x) + \mu \left( J * Z(t, x) - Z(t, x) \right) + \sigma W(t, x),$
where $\lambda, \mu \geq 0$ , $\sigma \in \mathbb{R}$ , and $W(t, x)$ represents time–space white noise interpreted in the distributional sense within the Hida space $(S)^*$ . The study focuses on the existence of mild solutions for singular (Dirac) initial data $Z(0, x) = \delta_0(x)$ .

The primary challenge addressed is the interplay between fractional time dynamics, nonlocal spatial interactions, and stochastic forcing. While purely local fractional stochastic heat equations have been studied (e.g., by Hachemi and Øksendal), the specific influence of the nonlocal term on the space–time integrability required for the existence of mild solutions in tempered distributions had not been fully clarified.

Methodology
The authors employ a combination of analytical and probabilistic techniques:

Transform Methods: The solution is derived using Laplace and Fourier transforms. The Laplace transform handles the Caputo time derivative, while the Fourier transform addresses the spatial operators.
Special Functions: The explicit solution formula relies heavily on Mittag–Leffler functions ( $E_\alpha$ and $E_{\alpha, \alpha}$ ), which serve as the natural analogues of the exponential function for fractional dynamics. The authors utilize sharp asymptotic expansions and bounds for these functions to analyze the behavior of the solution kernels.
Stochastic Analysis: The noise is treated via the Itô–Skorohod integral and Wick multiplication within the space of tempered distributions. The existence of a mild solution is reduced to verifying the finiteness of the second moment, $E[|Z_2(t, x)|^2] < \infty$ , where $Z_2$ is the stochastic convolution term.
Integrability Criteria: Using the Itô isometry and Plancherel's theorem, the problem of existence is transformed into a delicate space–time integrability problem involving the squares of Mittag–Leffler kernels. The authors establish optimal conditions by analyzing the asymptotic behavior of these kernels in both low and high frequencies.

Key Contributions and Results
The paper provides an explicit representation of the unique solution in the space of tempered distributions, decomposing it into a deterministic part ( $Z_1$ ) and a stochastic convolution part ( $Z_2$ ). The main contribution is a sharp characterization of the existence of mild solutions based on the parameters $(\alpha, N, \lambda, \mu)$ , where $N$ is the spatial dimension.

Theorem 1.2 establishes the following classification:

Purely Nonlocal Regime ( $\lambda = 0, \mu > 0$ ): No mild solution exists for any spatial dimension $N \geq 1$ and any $\alpha \in (0, 2)$ . The absence of the local Laplacian term prevents the necessary decay of the solution kernel at high frequencies.
Mixed/Local Dominant Regime ( $\lambda > 0, \mu \geq 0$ ): The existence of mild solutions depends critically on the relationship between $\alpha$ $α$ and $N$ $N$ :
- Case $\alpha = 1$ : A mild solution exists if and only if $N = 1$ .
- Case $0 < \alpha < 1$ : A mild solution exists if and only if $N = 1$ and $2/3 < \alpha < 1$ .
- Case $1 < \alpha < 2$ : A mild solution exists if and only if $N \in \{1, 2\}$ .

These results refine and extend previous findings for purely local equations (where $\mu=0$ ). Specifically, the authors correct a partial result in prior literature (Hachemi and Øksendal) by demonstrating that for $0 < \alpha < 1$ , mild solutions do exist in dimension $N=1$ provided the stricter condition $2/3 < \alpha < 1$ is met, whereas the previous work suggested non-existence for the entire range.

Significance and Claims
The paper claims to provide a complete classification of admissible regimes for this class of equations, highlighting the subtle role of fractional time derivatives.

Nonlocal vs. Local: A striking finding is that the nonlocal term alone cannot sustain a mild solution in the presence of white noise; the local Laplacian is essential for high-frequency regularization.
Memory Effects: The analysis confirms that the fractional derivative induces subdiffusive spreading and memory effects, which alter the integrability thresholds compared to classical stochastic heat equations.
Numerical Validation: The theoretical results are complemented by numerical simulations illustrating the evolution of the mean and variance. These simulations demonstrate the slower, subdiffusive spreading (scaling as $t^{\alpha/2}$ ) and the heavier tails characteristic of fractional dynamics compared to the classical Gaussian heat kernel.

The work is presented as an extension and sharpening of known results for purely local fractional stochastic heat equations, offering a rigorous framework for understanding systems with both memory and long-range spatial interactions.