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REVIEW 2 major objections 5 minor 32 references

Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The temporal process of a linear stochastic fractional diffusion equation is shown to satisfy Khinchin and Chung laws of the iterated logarithm at time zero, with explicit almost-sure constants at each fixed spatial point.

desk verdict The Khinchin half is solid and largely self-contained, but the Chung half rests on an asserted transfer of an exact small-ball theorem and needs referee pressure rather than desk rejection. read the letter →

arxiv 2608.04644 v1 pith:U7A4O2AK submitted 2026-08-05 math.PR

Chang Liu , Ran Wang This is my paper · ORCID
classification math.PR MSC 60H1560G1760G22
keywords stochasticfractionaldiffusionequationlawoftheiteratedlogarithmChung'ssmall-ballprobabilitiesself-similarGaussianprocessharmonizablerepresentationMittag-Lefflerfunctiontemporalregularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves sharp almost-sure oscillation rates at the initial time for the temporal process $t\mapsto u(t,x)$ of a broad class of linear stochastic fractional diffusion equations. For each fixed spatial point, Theorem 1.2 identifies the exact limsup constant $\tilde\kappa$, and Theorem 1.3 identifies the exact Chung-type liminf constant $\kappa\lambda_H^H$, where $H$ is the Hurst index of the Gaussian solution. These statements extend known time-zero laws for stochastic heat equations to time-fractional equations with Mittag-Leffler kernels and Riesz-type spatial noise. The initial time is genuinely different from positive times because the smoother remainder that is negligible away from zero contributes at the same order near $t=0$, so the proof needs a frequency-localization argument. The laws provide sharp benchmarks for nonlinear stochastic heat-type equations near time zero.

What carries the argument

The central object is the harmonizable representation of the solution, a Fourier-integral formula in which the value $u(t,x)$ is a stochastic integral over space-time frequencies and different frequency bands produce independent Gaussian processes. The argument decomposes the temporal process into a low-frequency and a high-frequency part, uses the self-similarity established in Lemma 2.1, and applies sharp frequency-truncation estimates from Lemma 2.2 and Proposition 2.2. For the Chung-type law, the load-bearing mechanism is the exact small-ball asymptotic of Proposition 4.1, transferred from a prior theorem after a Fourier-normalization check, together with a Gaussian comparison inequality and Borel-Cantelli localization over exponentially spaced time scales.

What would settle it

Evaluate Proposition 4.1 numerically for the parameter set $(\beta,\gamma,H_0,\alpha,\ell,d)=(1.5,0.6,0.5,1,1,1)$, which satisfies Condition 1.1, $H=0.85<1$, $0\le\gamma<1$, and $\beta+\gamma=2.1<2+H$: approximate $\varepsilon^{1/H}\log P(\sup_{0\le t\le 1}|u(t,x)|\le\varepsilon)$ for small $\varepsilon$ from many simulations of the harmonizable representation. If the values do not converge to $-\kappa^{1/H}\lambda_H$, the transferred small-ball theorem underlying Theorem 1.3 fails; if they do, the theorem's constant is corroborated.

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Extended reading notes

Core claim

The paper establishes two almost-sure laws for the temporal process $t\mapsto u(t,x)$ of the linear stochastic fractional diffusion equation, at the initial time. For each fixed $x$, this process is centered, Gaussian, and self-similar with Hurst index $H$ given by (1.5). Theorem 1.2 states that $\limsup_{t\downarrow 0}|u(t,x)|\big/\big(t^H\sqrt{2\log\log(1/t)}\big)=\tilde\kappa$ almost surely, and Theorem 1.3 states that, under $0\le\gamma<1$ and $\beta+\gamma<2+H$, $\liminf_{\varepsilon\downarrow 0}\sup_{0\le t\le\varepsilon}|u(t,x)|\big/\big(\varepsilon^H(\log\log\varepsilon^{-1})^{-H}\big)=\kappa\lambda_H^H$ almost surely. The central point is that both statements are sharp at time zero, where the smoother remainder of the pinned-string decomposition is no longer negligible; the proof controls that remainder through a harmonizable representation, frequency truncation, and localization.

Load-bearing premise

The Chung-type law's constant rests on a small-ball estimate taken from a prior theorem and assumed, after a Fourier-normalization check, to remain valid for the Riesz spectral density $|\xi|^{\ell-d}$ with the added restriction $\beta+\gamma<2+H$; if that transfer is wrong, Theorem 1.3 loses its constant while Theorem 1.2 would survive.

Editorial extensions

If this is right

  • For every fixed $x$, the pointwise temporal process satisfies $|u(t,x)|\sim\tilde\kappa\,t^H\sqrt{2\log\log(1/t)}$ in the limsup sense as $t\downarrow 0$, so the lowest-order growth at time zero is exactly of fractional-Brownian scale.
  • Under the extra conditions $0\le\gamma<1$ and $\beta+\gamma<2+H$, the running supremum over $[0,\varepsilon]$ is almost surely asymptotic to $\kappa\lambda_H^H\varepsilon^H(\log\log\varepsilon^{-1})^{-H}$, pinning the deepest small-time trough of the process.
  • The temporal process is self-similar with Hurst index $H$ at every fixed spatial point, and the almost-sure constants do not depend on $x$.
  • The Khinchin-type statement holds for all parameters in Condition 1.1 with $H<1$; the additional restrictions are needed only for the Chung-type statement.
  • These laws offer sharp initial-time benchmarks for nonlinear stochastic heat-type equations, where temporal increments are often compared with the linearized solution near $t=0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same frequency-block decomposition appears capable of producing exact small-time moduli of continuity for the temporal process, such as statements about $\sup_{s,t\in[0,\varepsilon]}|u(s)-u(t)|$, since the block estimates in Proposition 2.2 already control uniform fluctuations.
  • Beyond the paper, the Khinchin constant $\tilde\kappa$ is an explicit integral of Mittag-Leffler functions and could be evaluated numerically; Monte Carlo simulation of the harmonizable representation could independently test the value before relying on the small-ball transfer.
  • Beyond the paper, the boundary cases $H=1$ and $\gamma\ge 1$ remain open; if the exact small-ball asymptotic were extended to those regimes, the Chung-type constant would likely take a different form because the balance between the fractional-Brownian component and the smoother remainder changes.
  • Beyond the paper, for parameter ranges where $\beta+\gamma<2$, the remainder is automatically lower order and the Chung constant should be consistent with the stochastic-heat-equation limits already known for that family, providing a consistency check across parameter regimes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the linear stochastic fractional diffusion equation (1.1) driven by a Gaussian noise that is fractional in time and has Riesz-type spatial covariance. For each fixed spatial point x, the authors prove a Khinchin-type law of the iterated logarithm at time zero (Theorem 1.2, constant \tilde\kappa) and, under the additional hypotheses 0≤γ<1 and β+γ<2+H, a Chung-type law of the iterated logarithm (Theorem 1.3, constant κλ_H^H). The proofs rely on self-similarity of the temporal process (Lemma 2.1), sharp frequency-truncation estimates (Proposition 2.2), a covering-number argument for the Khinchin upper bound, a block-decomposition/Borel–Cantelli argument for the Khinchin lower bound, and an exact small-ball estimate (Proposition 4.1) transferred from the authors' earlier result [13, Theorem 4.3(1)].

Significance. If the results are correct, they provide sharp almost-sure initial-time oscillation rates with explicit constants for a broad family of time-fractional stochastic diffusion equations, extending previously known results for stochastic heat equations. The Khinchin part is the stronger contribution: it is self-contained apart from standard Gaussian concentration and covering estimates, and its constant \tilde\kappa is an explicit integral involving the Mittag–Leffler function. The Chung part, by contrast, hinges on Proposition 4.1, whose proof is a one-paragraph reduction to an external small-ball theorem; that transfer is not fully demonstrated. The paper uses no fitted parameters and states all constants explicitly, which is a definite strength.

major comments (2)
  1. [§4.1, Proposition 4.1 and Theorem 1.3] The proof of Proposition 4.1 is a reduction to [13, Theorem 4.3(1)] and does not actually demonstrate the transfer to the noise class (1.4). It asserts that the stationary-increment component has temporal spectral density I_{α,β,ℓ}/(2π)|τ|^{-2H-1}, that the smoother remainder has a strictly smaller small-deviation exponent under β+γ<2+H, and that the Gaussian correlation argument of [13] yields the same small-ball constant. However, no decomposition u(t,x)=κB_H(t)+R(t) is exhibited for the solution of (1.1), no bound on the small-deviation exponent of R is displayed, and the correlation between B_H and R is not controlled at the scale ε^{1/H}\log ε^{-1}. Since Theorem 1.3 rests entirely on (4.1), the constant κλ_H^H is unsupported unless Proposition 4.1 is proved in detail or the hypotheses of [13, Theorem 4.3(1)] are stated and verified for the Riesz-type density |ξ|^{ℓ-d}.
  2. [§1, Remark 1.1] Remark 1.1 acknowledges that the conditions 0≤γ<1 and β+γ<2+H are inherited from [13, Theorem 4.3(1)], but the paper does not state the precise hypotheses of that theorem. In particular, it is not specified which spatial covariance class [13] treats and whether the Riesz density |ξ|^{ℓ-d} with ℓ∈(0,2d∧2α) is included. Without this information the reader cannot verify that the reduction in Proposition 4.1 is legitimate, and the claim that the restriction β+γ<2+H is 'needed' remains an assertion. Please either state the referred theorem in full or provide a self-contained proof of (4.1).
minor comments (5)
  1. [Title/Abstract] The title and abstract contain spacing artifacts ('LA WS', 'ITERA TED', 'A T'); please correct them in the final version.
  2. [Eq. (1.7)–(1.8) and throughout] The constant \tilde\kappa is introduced in (1.8) with a tilde, but in several places (e.g., (1.7), Lemma 2.1, and the proof of Theorem 1.2) it is rendered as 'rκ'. Please use a consistent symbol.
  3. [§4.1] In the proof of Proposition 4.1, the phrase 'the temporal Fourier factor is (2π)^{-1}' is terse; a short derivation of the temporal spectral density from (1.4) and (2.11) would help the reader verify the normalization.
  4. [References] Reference [7] is an arXiv preprint from 2026; if a published version exists, please update the citation.
  5. [§2.2, Eq. (2.5)] The notation E_{a,b}(-x) is used without stating explicitly that the argument is real and non-positive; a sentence clarifying the real-valued convention would avoid ambiguity.

Circularity Check

1 steps flagged · score 4.0 of 10

Chung-type LIL rests on a self-cited exact small-ball theorem that is transferred to the Riesz spectral density by assertion rather than by proof; the Khinchin LIL is essentially self-contained.

  1. self citation load bearing [Section 4.1, Proposition 4.1 and Remark 1.1]
    "The exact small-ball result [13, Theorem 4.3(1)] is formulated for 0≤γ<1, which explains this hypothesis. We next verify the Fourier normalization and the correspondence between its parameters and those used here. ... The proofs of Proposition 4.2(1) and Theorem 4.3(1) in [13] use only the homogeneity of the spatial spectral measure, the Mittag–Leffler estimate (2.5), and the differentiability identity (2.6). The same estimates therefore apply to the Riesz-type density |ξ|^{ℓ−d}."

    Theorem 1.3, the Chung-type LIL, depends on the exact small-ball asymptotic stated in Proposition 4.1. The proof of Proposition 4.1 does not derive the small-ball constant for the Riesz spectral density; after identifying the main component with κB_H, it cites [13, Theorem 4.3(1)] and asserts that the estimates there carry over to |ξ|^{ℓ−d}. Reference [13] is co-authored by R. Wang, one of the present authors. The remainder control is also not exhibited: the condition β+γ<2+H is described as making the required exponent automatic, and Remark 1.1 states that the condition 0≤γ<1 is 'inherited' from [13, Theorem 4.3(1)]. Thus the central support for Theorem 1.3 reduces to a self-cited result plus a normalization check, rather than an independent argument.

full rationale

The paper contains no fitted-parameter or definitional circularity. Theorem 1.2 is proved from self-similarity, a Hölder estimate derived in the paper, Talagrand's Gaussian covering bound, and a frequency-block decomposition; no step reduces to its own conclusion. The serious issue is confined to Theorem 1.3. Its proof uses Proposition 4.1 as the exact small-ball input in both the lower and upper bounds. Proposition 4.1's proof is essentially a Fourier-normalization check followed by an assertion that the estimates of [13, Propositions 4.2(1) and Theorem 4.3(1)] carry over to the Riesz density |ξ|^{ℓ−d}. Since [13] has R. Wang as a co-author, this is a load-bearing self-citation: the paper does not independently prove the small-ball constant for the present noise model. The external constant λ_H for fractional Brownian motion is not the problem; the problem is the transfer of the exact small-ball result from [13] to the Riesz spectral density and the assertion about the smoother remainder. Remark 1.1 explicitly acknowledges that the extra hypotheses are inherited from [13]. This makes Theorem 1.3's support conditional on a self-cited result, but it is not a case of a prediction being forced by construction or by definition. A score of 4 reflects that the central Chung-type claim relies substantially on the authors' own previous theorem while the Khinchin claim is independent.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; all constants are explicit functionals of model parameters or universal constants. No new physical or mathematical entities such as particles, forces, dimensions, or conserved quantities are introduced. The harmonizable field v(A,t,x) and frequency-block processes v_n and v_tilde_n are auxiliary representations, not independent postulates.

assumptions (5)
  • domain assumption Existence and unique random field solution with Green function representation G(t-s,x-y) holds under Condition 1.1.
    Invoked at (1.6) via [7, Theorem 1.1(i)]; the present paper does not prove existence.
  • domain assumption The noise covariance (2.1) has the stated spectral representation with constant c_{H0}, and the stochastic integral extends to the completion of test functions.
    Specified in Section 2.1 and inherited from [7, p. 7]; needed for all covariance computations.
  • domain assumption Exact small-ball asymptotic for the solution is valid for the generalized Riesz-noise class, i.e. Proposition 4.1 follows from [13, Theorem 4.3(1)] under 0 <= gamma < 1 and beta + gamma < 2 + H.
    The proof of Proposition 4.1 is a normalization check plus an assertion that the arguments of [13] transfer; it is not carried out in this paper. This is the main external premise.
  • standard math Mittag-Leffler estimates: |E_{a,b}(-x)| <= C/(1+x) for x > 0, and the derivative identity (2.6).
    Used in Lemma 2.2 and Proposition 4.1; cited to [28, Theorem 1.6] and [7, (2.5)].
  • standard math Metric entropy tail bound for Gaussian processes [30, Theorem 2.4] and Gaussian isoperimetric inequality [11, Lemma 2.8].
    Used in Sections 3 and 4 for Borel-Cantelli estimates; standard external results.

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Pith. "Pith review of Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation." pith.science (2026). https://pith.science/paper/U7A4O2AK

@misc{pith2026260804644,
  author       = {Pith},
  title        = {Pith review of: Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7A4O2AK}},
  note         = {Machine review of arXiv:2608.04644}
}
abstract

We consider the linear stochastic fractional diffusion equation \begin{equation*} \partial^{\beta} u(t,x)=-\left(-\Delta\right)^{\alpha/2}u(t,x) +I_t^{\gamma}\bigl[\dot W(t,x)\bigr], \qquad t>0,\quad x\in\mathbb R^d, \end{equation*} with zero initial conditions, where $\alpha>0$, $\beta\in(0,2)$, and $\gamma\ge0$. The driving noise $\dot W$ is a centered Gaussian generalized field that is fractional in time and has Riesz-type spatial covariance. For each fixed $x\in\mathbb R^d$, we establish a Khinchin-type law of the iterated logarithm at time zero for the temporal process $t\mapsto u(t,x)$. Under the additional conditions $0\le\gamma<1$ and $\beta+\gamma<2+H$, we also prove the corresponding Chung-type law. The proofs rely on a harmonizable representation, sharp frequency-truncation estimates, an exact small-ball asymptotic, and a localization argument. These results extend the initial-time laws of the iterated logarithm for stochastic heat equations to a broad class of time-fractional stochastic diffusion equations.

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Works this paper leans on

32 extracted references · 31 canonical work pages

  1. [13]

    Y. H. Guo, J. Song, R. Wang and Y. Xiao, Sample path properties and small-ball probabilities for stochastic fractional diffusion equations,J. Differential Equations446(2025), paper no. 113604, 56 pp

    work page 2025

  2. [1]

    Balan and C

    R. Balan and C. Tudor, The stochastic heat equation with fractional-colored noise: existence of the solution,ALEA Lat. Am. J. Probab. Math. Stat.4(2008), 57–87

    work page 2008

  3. [2]

    Carfagnini and A

    M. Carfagnini and A. P. Todino, Small fluctuations for time-dependent spherical random fields, Electron. J. Probab.30(2025), paper no. 84, 1–25

    work page 2025

  4. [3]

    L. Chen, Y. Guo and J. Song, Moments and asymptotics for a class of SPDEs with space-time white noise,Trans. Amer. Math. Soc.377(2024), no. 6, 4255–4301

    work page 2024

  5. [4]

    L. Chen, Y. Hu and D. Nualart, Nonlinear stochastic time-fractional slow and fast diffusion equations onR d,Stochastic Process. Appl.129(2019), no. 12, 5073–5112

    work page 2019

  6. [5]

    L. Chen, J. Huang, D. Khoshnevisan and K. Kim, Dense blowup for parabolic SPDEs,Electron. J. Probab.24(2019), 1–33

    work page 2019

  7. [6]

    Chen and K

    L. Chen and K. Kim, Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency,Acta Math. Sci. Ser. B (Engl. Ed.)39(2019), no. 3, 645–668

    work page 2019

  8. [7]

    L. Chen, C. Y. Lee and P. Xia, Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations, arXiv:2602.05317 (2026)

    work page arXiv 2026

Show all 32 references
  1. [8]

    R. C. Dalang, D. Nualart and F. Pu, Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line. Preprint available at arXiv:2508.11859 (2025)

    2025 arXiv

  2. [9]

    R. C. Dalang and M. Sanz-Sol´ e,Stochastic Partial Differential Equations, Space-Time White Noise and Random Fields, Springer Monographs in Mathematics, Springer, Cham, 2026

    2026

  3. [10]

    Das, Temporal increments of the KPZ equation with general initial data.Electron

    S. Das, Temporal increments of the KPZ equation with general initial data.Electron. J. Probab. 29(2024), 1–28. KHINCHIN’S AND CHUNG’S LILS AT TIME ZERO FOR AN SFDE 21

    2024

  4. [11]

    N. D. Feldheim, O. N. Feldheim and L. P. Lakshmi, A sharp transition in zero overcrowding and undercrowding probabilities for stationary Gaussian processes,J. Anal. Math.158(2026), no. 1, 211–232

    2026

  5. [12]

    Foondun, D

    M. Foondun, D. Khoshnevisan and P. Mahboubi, Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion.Stoch PDE: Anal. Comp.3(2015), 133–158

    2015

  6. [14]

    Hairer and ´E

    M. Hairer and ´E. Pardoux, A Wong–Zakai theorem for stochastic PDEs,J. Math. Soc. Japan 67(2015), no. 4, 1551–1604

    2015

  7. [15]

    Herrell, R

    R. Herrell, R. Song, D. Wu and Y. Xiao, Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise,Stoch. Anal. Appl.38(2020), no. 4, 747–768

    2020

  8. [16]

    Huang and D

    J. Huang and D. Khoshnevisan, On the multifractal local behavior of parabolic stochastic PDEs.Electron. Commun. Probab.22(2017), 1–11

    2017

  9. [17]

    Z. M. Khalil and C. A. Tudor, On the distribution andq-variation of the solution to the heat equation with fractional Laplacian,Probab. Math. Statist.39(2019), no. 2, 315–335

    2019

  10. [18]

    Khoshnevisan,Analysis of Stochastic Partial Differential Equations, CBMS Regional Con- ference Series in Mathematics, Vol

    D. Khoshnevisan,Analysis of Stochastic Partial Differential Equations, CBMS Regional Con- ference Series in Mathematics, Vol. 119, American Mathematical Society, Providence, RI, 2014

    2014

  11. [19]

    Khoshnevisan, K

    D. Khoshnevisan, K. Kim and C. Mueller, Small-ball constants, and exceptional flat points of SPDEs.Electron. J. Probab.29(2024), paper no. 180, 31 pp

    2024

  12. [20]

    Khoshnevisan, J

    D. Khoshnevisan, J. Swanson, Y. Xiao and L. Zhang, Weak existence of a solution to a differ- ential equation driven by a very rough fBm. arXiv:1309.3613v2 (2014)

    2014 arXiv

  13. [21]

    Kilbas and M

    A. Kilbas and M. Saigo,H-Transforms: Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL, 2004

    2004

  14. [22]

    Lee and Y

    C. Lee and Y. Xiao, Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields,Bernoulli29(2023), no. 1, 523–550

    2023

  15. [23]

    Lei and D

    P. Lei and D. Nualart, A decomposition of the bifractional Brownian motion and some appli- cations,Statist. Probab. Lett.79(2009), no. 5, 619–624

    2009

  16. [24]

    W. V. Li and Q. M. Shao, Gaussian processes: Inequalities, small-ball probabilities and appli- cations, inStochastic Processes: Theory and Methods, Handbook of Statist., Vol. 19 (North- Holland, Amsterdam, 2001), pp. 533–597

    2001

  17. [25]

    Liu and R

    C. Liu and R. Wang, Chung’s LIL for the linear stochastic fractional heat equation at origin, Statist. Probab. Lett.235(2026), paper no. 110737

    2026

  18. [26]

    Mueller and R

    C. Mueller and R. Tribe, Hitting properties of a random string,Electron. J. Probab.7(2002), paper no. 10, 29 pp

    2002

  19. [27]

    Olver, D

    F. Olver, D. Lozier, R. Boisvert and C. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC/Cambridge, 2010

    2010

  20. [28]

    I. Podlubny,Fractional Differential Equations: An Introduction to Fractional Derivatives, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, CA, 1999

    1999

  21. [29]

    B. Qian, M. Wang, R. Wang and Y. Xiao, Temporal regularity for the nonlinear stochastic heat equation with spatially rough noise,J. Differential Equations461(2026), paper no. 114097, 41 pp

    2026

  22. [30]

    Talagrand, Sharper bounds for Gaussian and empirical processes,Ann

    M. Talagrand, Sharper bounds for Gaussian and empirical processes,Ann. Probab.22(1994), no. 1, 28–76

    1994

  23. [31]

    C. A. Tudor and Y. Xiao, Sample path properties of the solution to the fractional-colored stochastic heat equation,Stoch. Dyn.17(2017), no. 1, 1750004. 22 C. LIU AND R. W ANG

    2017

  24. [32]

    Wang and S

    R. Wang and S. Zhang, Decompositions of stochastic convolution driven by a white-fractional Gaussian noise,Front. Math. China16(2021), no. 4, 1063–1073. Chang Liu, School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. Email address:changliu0504@163.com R...

    2021

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