Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient
Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-th...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-11-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/11/815 |
| _version_ | 1797459284230930432 |
|---|---|
| author | Wensheng Wang |
| author_facet | Wensheng Wang |
| author_sort | Wensheng Wang |
| collection | DOAJ |
| description | Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set <i>B</i>’s packing dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo form="prefix">dim</mo><msub><mrow></mrow><mi>p</mi></msub></msub><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients. |
| first_indexed | 2024-03-09T16:49:15Z |
| format | Article |
| id | doaj.art-c7d9e789e67a403c8d79eed5ed9125ba |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:49:15Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-c7d9e789e67a403c8d79eed5ed9125ba2023-11-24T14:43:04ZengMDPI AGFractal and Fractional2504-31102023-11-0171181510.3390/fractalfract7110815Temporal Fractal Nature of the Time-Fractional SPIDEs and Their GradientWensheng Wang0School of Economics, Hangzhou Dianzi University, Hangzhou 310018, ChinaFractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set <i>B</i>’s packing dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo form="prefix">dim</mo><msub><mrow></mrow><mi>p</mi></msub></msub><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.https://www.mdpi.com/2504-3110/7/11/815TFSPIDEsBrownian-time processesspace–time white noisetemporal fractal naturehitting probabilitiesHölder regularity |
| spellingShingle | Wensheng Wang Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient Fractal and Fractional TFSPIDEs Brownian-time processes space–time white noise temporal fractal nature hitting probabilities Hölder regularity |
| title | Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient |
| title_full | Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient |
| title_fullStr | Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient |
| title_full_unstemmed | Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient |
| title_short | Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient |
| title_sort | temporal fractal nature of the time fractional spides and their gradient |
| topic | TFSPIDEs Brownian-time processes space–time white noise temporal fractal nature hitting probabilities Hölder regularity |
| url | https://www.mdpi.com/2504-3110/7/11/815 |
| work_keys_str_mv | AT wenshengwang temporalfractalnatureofthetimefractionalspidesandtheirgradient |