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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.15427 |
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| _version_ | 1866910309949636608 |
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| author | Bouafia, Philippe De Pauw, Thierry |
| author_facet | Bouafia, Philippe De Pauw, Thierry |
| contents | A function $f$ defined on $[0, 1]^d$ is called strongly chargeable if there is a continuous vector-field $v$ such that $f(x_1, \dots,x_d)$ equals the flux of $v$ through the rectangle $[0, x_1] \times \cdots \times [0, x_d]$ for all $(x_1, \dots, x_d) \in [0, 1]^d$. In other words, $f$ is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with $d \geq 2$ parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter $(H_1, \dots, H_d)$ are shown to be almost surely strongly chargeable whenever
\[
\frac{H_1 + \cdots + H_d}{d} > \frac{d - 1}{d}.
\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A regularity property of fractional Brownian sheets Bouafia, Philippe De Pauw, Thierry Probability Analysis of PDEs 60G22, 60G17, 26A45 A function $f$ defined on $[0, 1]^d$ is called strongly chargeable if there is a continuous vector-field $v$ such that $f(x_1, \dots,x_d)$ equals the flux of $v$ through the rectangle $[0, x_1] \times \cdots \times [0, x_d]$ for all $(x_1, \dots, x_d) \in [0, 1]^d$. In other words, $f$ is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with $d \geq 2$ parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter $(H_1, \dots, H_d)$ are shown to be almost surely strongly chargeable whenever \[ \frac{H_1 + \cdots + H_d}{d} > \frac{d - 1}{d}. \] |
| title | A regularity property of fractional Brownian sheets |
| topic | Probability Analysis of PDEs 60G22, 60G17, 26A45 |
| url | https://arxiv.org/abs/2401.15427 |