# Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient

## Abstract

**:**

## 1. Introduction

**Theorem 1.**

**Remark 1.**

- Equations (5) and (7) are other forms of the global temporal continuity moduli of the TFSPIDEs and the TFSPIDE gradients, respectively, which are slightly different from those obtained in [13]. Equation (5) with ${k}_{8}^{(\beta ,d)}{|t-s|}^{\frac{2-d\beta}{4}}\sqrt{2log(1/|t-s\left|\right)}$ taking the place of ${\varphi}_{\beta ,d,h}$, and Equation (7) with ${k}_{12}{|t-s|}^{\frac{2-3\beta}{4}}$$\sqrt{log(1/|t-s\left|\right)}$ taking the place of ${\phi}_{\beta ,h}$ were established in [13], where ${k}_{8}^{(\beta ,d)}>0$ and ${k}_{12}>0$ were understood as dimension-dependent constants, i.e., independent of x (whose exact values were unknown). Here, in Equations (5) and (7), we give the exact constants for the global temporal continuity moduli of the TFSPIDEs and the TFSPIDE gradients. Moreover, by using Lemma 5 below, we can obtain ${k}_{8}^{(\beta ,d)}=\sqrt{2{K}_{\beta ,d}}$ and ${k}_{12}=\sqrt{2{K}_{\beta ,0}}$, as was obtained in [13]. In this sense, the results of this paper reinforce those in [13].
- Equation (6) with ${k}_{9}^{(\beta ,d)}{h}^{\frac{2-d\beta}{4}}\sqrt{loglog(1/h)}$ taking the place of ${\widehat{\varphi}}_{\beta ,d,h}$, and Equation (8) with ${k}_{13}{h}^{\frac{2-3\beta}{4}}\sqrt{loglog(1/h)}$ taking the place of ${\widehat{\phi}}_{\beta ,h}$ were established in [13], where ${k}_{9}^{(\beta ,d)}>0$ and ${k}_{13}>0$ were understood as dimension-dependent constants, i.e., independent of x (whose exact values were unknown). Here, in Equations (6) and (8), we give the exact constants for the temporal LILs of the TFSPIDEs and the TFSPIDE gradients. Moreover, by using Lemma 5 below, we can obtain ${k}_{9}^{(\beta ,d)}=\sqrt{2{K}_{\beta ,d}}$ and ${k}_{13}=\sqrt{2{K}_{\beta ,0}}$, as was obtained in [13]. In this sense, the results of this paper reinforce those in [13].
- Equation (5) gives the magnitude of the global maximal oscillation of the TFSPIDE solution ${U}_{\beta}(\xb7,x)$ over the compact rectangle ${I}_{\mathrm{time}}$, which is ${\varphi}_{\beta ,d,h}$. Equation (7) gives the magnitude of the global maximal oscillation of the TFSPIDE gradient solution ${\partial}_{x}{U}_{\beta}(\xb7,x)$ over the compact rectangle ${I}_{\mathrm{time}}$, which is ${\phi}_{\beta ,h}$.
- Equation (6) gives the magnitude of the local oscillation of the TFSPIDE solution ${U}_{\beta}(\xb7,x)$ at a prescribed time ${t}_{0}\ge 0$ is ${\widehat{\varphi}}_{\beta ,d,h}$. Equation (8) gives the magnitude of the local oscillation of the TFSPIDE gradient solution ${\partial}_{x}{U}_{\beta}(\xb7,x)$ at a prescribed time ${t}_{0}\ge 0$ is ${\widehat{\phi}}_{\beta ,h}$.
- It is interesting to compare Equations (5) and (6). The latter one states that, at some given point, the LIL of ${U}_{\beta}(\xb7,x)$ for any fixed x is not more than ${\widehat{\varphi}}_{\beta ,d,h}$. On the other hand, the former tells us that the global continuity modulus of ${U}_{\beta}(\xb7,x)$ can be much larger, namely ${\varphi}_{\beta ,d,h}$. Similarly, by Equations (7) and (8), the LIL of ${\partial}_{x}{U}_{\beta}(\xb7,x)$ for every fixed x is less than ${\widehat{\phi}}_{\beta ,h}$. On the other hand, the continuity modulus of ${\partial}_{x}{U}_{\beta}(\xb7,x)$ can be much larger, namely ${\phi}_{\beta ,h}$.
- With Equation (6) and Fubini’s theorem, we have the random time set at$${S}_{\beta ,d,x,+}:=\left\{t\in [0,1]:\underset{h\to 0+}{lim\; sup}{\widehat{\varphi}}_{\beta ,d,h}^{-1}|{U}_{\beta}(t+h,x)-{U}_{\beta}(t,x)|>1\right\},$$

**Theorem 2.**

**Theorem 3.**

**Remark 2.**

**Remark 3.**

## 2. Preliminaries

#### 2.1. Rigorous Kernel SIE Formulations

**Lemma 1**

#### 2.2. Estimations on the Variances of Temporal Increments of TFSPIDEs and Their Gradients

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Proof.**

## 3. Results

#### 3.1. Temporal Moduli of Continuity

**Lemma 4.**

**Proof.**

**Lemma 5.**

**Proof.**

#### 3.2. Hausdorff Dimensions for the Sets of Temporal Fast Points

**Proof of Theorem 2.**

**Lemma 6.**

**Lemma 7.**

**Proof.**

**Lemma 8.**

**Proof.**

**Lemma 9.**

**Proof.**

**Lemma 10.**

**Proof.**

#### 3.3. Hitting Probabilities for the Sets of Temporal Fast Points

**Proof of Theorem 3.**

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TFSPIDE | Time-fractional stochastic partial integro-differential equation |

ISLTBM | Inverse-stable-Lévy-time Brownian motion |

PDE | Partial differential equation |

BTBM | Brownian-time Brownian motion |

BTP | Brownian-time process |

SIE | Stochastic integral equation |

LIL | Law of the iterated logarithm |

## References

- Allouba, H.; Nane, E. Interacting time-fractional and Δ
^{v}PDEs systems via Brownian-time and Inverse-stable-Lévy-time Brownian sheets. Stoch. Dynam.**2013**, 13, 1250012. [Google Scholar] [CrossRef] - Caputo, M. Linear models of dissipation whose Q is almost frequency independent. Part II Geophys. J. Roy. Astr. Soc.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Srivastava, H.M.; Adel, W.; Izadi, M.; El-Sayed, A.A. Solving some physics problems involving fractional-order differential equations with the morgan-voyce Polynomials. Fractal Fract.
**2023**, 7, 301. [Google Scholar] [CrossRef] - Chalishajar, D.; Kasinathan, R.; Kasinathan, R. Optimal control for neutral stochastic integrodifferential equations with infinite delay driven by Poisson jumps and rosenblatt process. Fractal Fract.
**2023**, 7, 783. [Google Scholar] [CrossRef] - D’Ovidio, M.; Orsingher, E.; Toaldo, B. Time-changed processes governed by space-time fractional telegraph equations. Stoch. Anal. Appl.
**2014**, 32, 1009–1045. [Google Scholar] [CrossRef] - Garra, R.; Orsingher, E.; Polito, F. Fractional diffusions with time-varying coefficients. J. Math. Phys.
**2015**, 56, 093301. [Google Scholar] [CrossRef] - Chen, Z.-Q.; Kim, K.-H.; Kim, P. Fractional time stochastic partial differential equations. Stoch. Proc. Appl.
**2015**, 125, 1470–1499. [Google Scholar] [CrossRef] - Mijena, J.B.; Erkan, N. Space-time fractional stochastic partial differential equations. Stoch. Proc. Appl.
**2015**, 125, 3301–3326. [Google Scholar] [CrossRef] - Allouba, H. A Brownian-time excursion into fourth-order PDEs, linearized Kuramoto-Sivashinsky, and BTPSPDEs on ${\mathbb{R}}_{+}\times {\mathbb{R}}^{d}$. Stoch. Dynam.
**2006**, 6, 521–534. [Google Scholar] [CrossRef] - Allouba, H. Time-fractional and memoryful ${{\Delta}^{2}}^{k}$ SIEs on ${\mathbb{R}}_{+}\times {\mathbb{R}}^{d}$: How far can we push white noise? Ill. J. Math.
**2013**, 57, 919–963. [Google Scholar] - Allouba, H. Brownian-time Brownian motion SIEs on ${\mathbb{R}}_{+}\times {\mathbb{R}}^{d}$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links. Discrete Contin. Dyn. Syst.
**2013**, 33, 413–463. [Google Scholar] [CrossRef] - Allouba, H. L-Kuramoto-Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp Hölder regularity, and Swift-Hohenberg law equivalence. J. Differ. Equ.
**2015**, 259, 6851–6884. [Google Scholar] [CrossRef] - Allouba, H.; Xiao, Y. L-Kuramoto-Sivashinsky SPDEs v.s. time-fractional SPIDEs: Exact continuity and gradient moduli, 1/2-derivative criticality, and laws. J. Differ. Equ.
**2017**, 263, 15521610. [Google Scholar] [CrossRef] - Wang, W. Spatial moduli of non-differentiability for time-fractional SPIDEs and their gradient. Symmetry
**2021**, 13, 380. [Google Scholar] [CrossRef] - Wang, W. Variations of the solution to a fourth order time-fractional stochastic partial integro-differential equation. Stoch. Partial Differ.
**2022**, 10, 582–613. [Google Scholar] [CrossRef] - Orey, S.; Taylor, S.T. How often on a Brownian path does the iterated logarithm fail? P. Lond. Math. Soc.
**1974**, 28, 174–192. [Google Scholar] [CrossRef] - Deheuvels, P.; Mason, P. On the fractal nature of empirical increments. Ann. Probab.
**1995**, 23, 355–387. [Google Scholar] [CrossRef] - Zhang, L.-X. On the fractal nature of increments of ℓ
^{p}-valued Gaussian processes. Stoch. Proc. Appl.**1997**, 71, 91–110. [Google Scholar] [CrossRef] - Khoshnevisan, D.; Peres, Y.; Xiao, Y. Limsup random fractals. Electron J. Probab.
**2000**, 5, 1–24. [Google Scholar] [CrossRef] - Falconer, K.J. The Geometry of Fractal Sets; Cambridge Univ. Press: Cambridge, UK, 1985. [Google Scholar]
- Taylor, S.J. The measure theory of random fractals. Math. Proc. Camb. Phil. Soc.
**1986**, 100, 383–406. [Google Scholar] [CrossRef] - Mattila, P. Geometry of Sets and Measures in Euclidean Spaces; Cambridge Univ. Press: Cambridge, UK, 1995. [Google Scholar]
- Mueller, C.; Tribe, R. Hitting probabilities of a random string, Electron. J. Probab.
**2002**, 7, 10–29. [Google Scholar] - Tudor, C.A. Analysis of Variations for Self-Similar Processes—A Stochastic Calculus Approach; Springer: Cham, Switzerland, 2013. [Google Scholar]
- Tudor, C.A.; Xiao, Y. Sample path properties of the solution to the fractional-colored stochastic heat equation. Stoch. Dynam.
**2017**, 17, 1750004. [Google Scholar] [CrossRef] - Fang, K.T.; Kotz, S.; Ng, K.W. Symmetric Multivariate and Related Distribution; Chapman and Hall Ltd.: London, UK, 1990. [Google Scholar]
- Haubold, H.J.; Mathai, A.M.; Saxena, R.K. Mittag-Leffler Functions and Their Applications. Hindawi Publishing Corporation. J. Appl. Math.
**2011**, 2011, 298628. [Google Scholar] [CrossRef] - Csörgo, M.; Révész, P. Strong Approxiamtions in Probability and Statistics; Academic Press: New York, NY, USA, 1981. [Google Scholar]
- Meerschaert, M.M.; Wang, W.; Xiao, Y. Fernique type inequality and moduli of continuity for anisotropic Gaussian random fields. Trans. Am. Math. Soc.
**2013**, 365, 1081–1107. [Google Scholar] [CrossRef] [PubMed] - Ledoux, M.; Talagrand, M. Probability in Banach Spaces; Springer: Berlin, Germany, 1991. [Google Scholar]
- Joyce, H.; Preiss, D. On the existence of subsets of finite positive packing measure. Mathematika
**1995**, 42, 15–24. [Google Scholar] [CrossRef] - Munkres, J.R. Topology: A First Course; Prentice-Hall Inc.: Englewood Cliffs, NJ, USA, 1975. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, W.
Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient. *Fractal Fract.* **2023**, *7*, 815.
https://doi.org/10.3390/fractalfract7110815

**AMA Style**

Wang W.
Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient. *Fractal and Fractional*. 2023; 7(11):815.
https://doi.org/10.3390/fractalfract7110815

**Chicago/Turabian Style**

Wang, Wensheng.
2023. "Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient" *Fractal and Fractional* 7, no. 11: 815.
https://doi.org/10.3390/fractalfract7110815