Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise

Jincheng Dong; Ning Du; Xu Guo; Mengmeng Liu; Xiangcheng Zheng

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Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise

Jincheng Dong, Ning Du, Xu Guo, Mengmeng Liu, Xiangcheng Zheng

arXiv:2608.29533v1Aug 30, 2026math.NA

TL;DR

Stochastic multiscale subdiffusion combines a variable-exponent Abel kernel with fractional Gaussian noise, creating kernel-analysis and low-regularity challenges. The paper proves well-posedness and regularity, develops semidiscrete and fully discrete schemes, and validates their theoretical convergence behavior numerically.

Problem
Variable-exponent multiscale kernels lack favorable analytical properties, while fractional Gaussian noise produces low regularity that complicates analysis and numerical approximation.
Method
The paper uses a perturbation technique and solution-operator approach to analyze mild solutions, then develops semidiscrete-in-time and fully discrete schemes.
Results
The paper proves unique mild-solution well-posedness, spatial and temporal regularity, and temporal convergence rates for the fully discrete scheme consistent with Theorem 6.1.
Takeaways & Limitations
The results provide a theoretical and numerical analysis of stochastic multiscale subdiffusion, with experiments verifying the sharpness of the error estimate.
Takeaways & Limitations
The analysis assumes α′(t) and α′′(t) exist and remain bounded on [0,T].

Abstract

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This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $α(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.

1. Introduction.

The paper studies stochastic multiscale subdiffusion with fractional Gaussian noise, replacing a constant exponent by α(t) to represent transitions between diffusive regimes. It addresses the multiscale kernel’s analytical difficulties and noise-induced low regularity by proving solution properties and developing time- and space-discrete schemes.

Motivation: Variable α(t) in the multiscale Abel kernel models transitions between diffusive regimes that single-scale subdiffusion cannot properly depict.The kernel is t^α(t)−1/Γ(α(t)); experimental comparisons reported in the introduction indicate better description with multiscale subdiffusion.
Model: The model is driven by fractional Gaussian noise, one of several stochastic noises used in subdiffusion studies.The paper describes fractional Gaussian noise as the derivative of fractional Brownian motion.
Assumptions: The exponent satisfies α(t)∈(0,1), with bounded α′(t) and α′′(t) assumed on [0,T].These assumptions imply continuity of α and a positive lower bound α* over the time interval.
Analytical challenges: The multiscale Abel kernel lacks complete monotonicity and an explicit integral-transform representation, so single-scale analysis tools do not apply directly.The paper therefore uses a perturbation method to handle the variable exponent.
Contributions: The paper proves well-posedness and solution regularity, then develops and analyzes semidiscrete-in-time and fully discrete numerical schemes.The analysis includes spatial and temporal regularity estimates and numerical error analysis.

2. Modeling issues and well-posedness.

The paper reformulates the stochastic multiscale problem using functional-analytic preliminaries, a kernel perturbation, and mild-solution operators. A contraction argument then establishes existence and uniqueness under the stated assumptions.

Preliminaries: The analysis introduces Sobolev, Hilbert, covariance, and stochastic-integral spaces to formulate the fractional-noise problem.Fractional Brownian motion uses covariance operator Q and independent one-dimensional motions with Hurst index H∈(1/2,1).
Model reformulation: The model is decomposed into deterministic and stochastic problems and represented through mild solutions based on the Laplace-transform solution operator.The mild solution combines the initial-data contribution, fractional Gaussian noise, and a kernel-perturbation term.
Model reformulation: The variable-exponent kernel is split using a perturbation method, with remainder terms controlled through bounds on g and g′.The estimates include |g|≤Ct^α0(1+|ln t|) and |g′|≤Ct^α0−1−ε.
Well-posedness: For sufficiently large ς satisfying Cς^−1+ε<1, the solution mapping is a contraction on X, yielding existence and uniqueness of a mild solution.The mapping is first shown to be well defined and then contractive using convolution estimates.
Well-posedness: The well-posedness result assumes the stated operator regularity condition and G0∈H, and places the unique mild solution in X.The contraction proof provides the existence and uniqueness conclusion for the original stochastic problem.

3. Solution regularity.

The paper derives spatial and temporal regularity estimates for the mild solution by separating deterministic and stochastic components. These estimates support subsequent low-regularity numerical analysis and are consistent with the constant-exponent case when α(t) is constant.

Regularity framework: The solution is decomposed into deterministic and stochastic components, whose regularity is analyzed separately before combining the estimates.The decomposition is G=u+v, where u solves the stochastic problem and v solves the deterministic problem.
Stochastic regularity: For the stochastic component, temporal Hölder regularity holds with exponent γ∈(0,H−ρα0) under ρ∈[0,H/α0)∩[0,1].The estimate bounds the mean-square temporal increment after normalization by τ^γ.
Deterministic regularity: The deterministic component satisfies spatial regularity estimates depending on the initial-data regularity and the parameters σ, q, and α0.For G0∈Ĥ^2q(D), q∈[0,σ], the estimate includes the factor t^(2q−2σ)α0.
Original problem: The full mild solution inherits spatial regularity from the component estimates and temporal Hölder regularity from the stochastic and deterministic bounds.The paper explicitly derives the original problem’s spatial estimate and combines component results for the full temporal estimate.
Original problem: When α(t)≡α, the spatial and temporal regularity results agree with the corresponding constant-exponent results.This consistency is stated as a direct comparison with the constant-exponent case.

4. Temporal discretization.

The paper constructs semidiscrete-in-time schemes using backward Euler and convolution quadrature, with coefficient decompositions and bounds tailored to the variable exponent α(t).

The variable-exponent kernel is represented through auxiliary functions H(t) and U(z,t) that incorporate α′(t) and logarithmic terms.
The discretization uses truncated auxiliary quantities Ȟn,k(s) and Ǔn,k(z,s), with truncation errors Jn and Ĵn defined explicitly.
The convolution coefficients are decomposed as bn,k = b̂n,k + b̃n,k to separate their components for analysis.
The temporal schemes combine the discretization of g′∗Au with backward Euler and convolution quadrature.
The semidiscrete approximation to the stochastic equation is defined by combining the two component solutions as Gn := un + vn.
If α(t) is bounded below by α* > 0 and |α′(t)| ≤ L, the coefficients satisfy bounds uniform in n, k, and τ.

5. Auxiliary estimates.

This section develops integral representations and regularity estimates for the semidiscrete solutions, establishing the auxiliary bounds needed for later error analysis.

Integral representations of the solutions to schemes (4.3) and (4.5) are derived to support spatial regularity and error estimates.
The temporal discretization analysis uses contour-based solution representations and auxiliary operators defined through Laplace-transform quantities.
The discrete solution of scheme (4.3) satisfies a spatial regularity estimate controlled by the time level and the exponent α0.
The analysis establishes bounds for auxiliary convolution terms by combining operator estimates, contour representations, and regularity lemmas.
For scheme (4.5), the regularity estimate has the form ∥A^σvn∥ ≤ Ct_n^-σα0.

6. Error estimate of semi-discrete approximations.

The paper derives temporal error estimates for the semidiscrete approximations by comparing continuous and discrete solution operators and controlling the effects of fractional Gaussian noise and variable exponents.

The main semidiscrete result provides an error estimate for the approximation Gn to the stochastic multiscale subdiffusion model.
The stochastic contribution is controlled using fractional Gaussian noise estimates involving the Hurst parameter H and spatial regularity exponent ρ.
The proof introduces the auxiliary operator Eτ(z) to measure the difference between continuous and discrete resolvent representations.
Under bounded α′(t) and α′′(t), the remaining terms in the error estimate are bounded by Cτ^2 when t_n − s ≤ 1.
The analysis bounds operator differences by decomposing them into continuous-time variation and discretization-error terms.
For the second semidiscrete component, the reported error estimate has the form ∥v(t_n) − vn∥ ≤ Cτ^(1−ε) + Ct_n^-1.

7. Full discretization and error estimate.

The fully discrete method combines the temporal schemes with piecewise linear finite elements, and the paper derives spatial and overall error estimates for both components.

Finite element discretization: The fully discrete schemes use finite element spaces of continuous piecewise linear functions with homogeneous boundary conditions.
Finite element discretization: The spatial discretization introduces the L2 projection Ph and Ritz projection Rh, together with a Ritz projection error estimate.
Fully discrete schemes: The fully discrete solutions admit contour-integral representations analogous to those used for the semidiscrete schemes.
Spatial error estimates: A finite element resolvent estimate gives an h^(2−2s) spatial factor for s ∈ [0,1/2].
Spatial error estimates: The paper derives separate spatial error estimates for the fully discrete schemes corresponding to the two components.
Overall error estimate: The main fully discrete theorem combines the componentwise estimates to obtain an overall error estimate for the approximation Gn.

8. Numerical experiments.

The numerical experiments examine temporal and spatial convergence of the fully discrete schemes under varied kernel exponents, Hurst indices, and noise regularities. They report agreement with theoretical estimates and illustrate how fractional Gaussian noise affects solution trajectories.

Experimental setup: The experiments vary (α0, αT), H, and m to assess temporal and spatial convergence of the proposed fully discrete schemes.The temporal studies use α-pairs (0.3, 0.7), (0.5, 0.2), and (0.8, 0.5), with H = 0.6 or 0.75 and m = 0, −0.5, or −1.
Temporal convergence: For m = 0 and ρ > 1/4, the temporal convergence order is O(τ^(H−α0/4)), as observed in Tables 1 and 2.The reported temporal behavior is tied to the baseline noise-regularity setting m = 0.
Temporal convergence: Increasing noise regularity with m = −0.5 or m = −1 improves the observed temporal convergence rates, consistent with Theorem 6.1.These settings correspond respectively to ρ > 1/8 and ρ > 0.
Sample-path behavior: Larger H produces smoother solution trajectories, whereas smaller H produces stronger fluctuations; ensemble averaging smooths sample-induced variability.The comparison uses H = 0.6, 0.75, and 0.9, with 100 independent realizations for the ensemble mean.

9. Conclusion.

The paper studies multiscale subdiffusion driven by fractional Gaussian noise, proves well-posedness and regularity, and develops a fully discrete numerical scheme whose error-estimate sharpness is verified experimentally.

The model describes anomalous diffusion with time-varying sublinear mean squared displacement growth and noise with autocorrelation structure.
Table 8 reports spatial errors and observed convergence rates for schemes (4.3) and (4.5) with H = 0.75, m = 0, and ρ > 1/4.
Tables 9 and 10 report spatial errors and observed convergence rates for schemes (4.3) and (4.5) with H = 0.6 or 0.75, m = −0.5, and ρ > 1/8.
The paper develops and analyzes a fully discrete numerical scheme for the stochastic multiscale subdiffusion problem.
Extensive numerical experiments verify the sharpness of the error estimate.