Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.10936 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914605709656064 |
|---|---|
| author | Grothaus, Martin da Silva, Jose Luis Suryawan, Herry Pribawanto Ullrich, Thomas |
| author_facet | Grothaus, Martin da Silva, Jose Luis Suryawan, Herry Pribawanto Ullrich, Thomas |
| contents | By using white noise analysis, we study the integral kernel $ξ(x)$, $x\in\mathbb{R}^{d}$, of stochastic currents corresponding to fractional Brownian motion with Hurst parameter $H\in(0,1)$. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and $d\ge1$ we show that the kernel $ξ(x)$ is well-defined as a Hida distribution for all $H\in(0,1)$. For $x=0$ and $d=1$, $ξ(0)$ is a Hida distribution for all $H\in(0,1)$. For $d\ge2$, then $ξ(0)$ is a Hida distribution only for $H\in(0,1/d)$. For $d=1$, $x \neq 0$, and $H \in (0,1)$, we show that $ξ(x) \in \mathcal{G}'$, the space of regular generalized functions. Elements of the space $\mathcal{G}'$ and elements from the negative Sobolev--Watanabe distribution spaces share the property that partial sums of their chaos decomposition are square integrable functions. More precisely, we show that $ξ(x) \in \mathcal{G}_{-s} \subset \mathcal{G}'$ for $x \neq 0$, $H \in (0,1)$, and all $s > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Currents of Fractional Brownian Motion: Existence and Regularity Grothaus, Martin da Silva, Jose Luis Suryawan, Herry Pribawanto Ullrich, Thomas Probability 60H40, 60J65, 60G22, 46F25 By using white noise analysis, we study the integral kernel $ξ(x)$, $x\in\mathbb{R}^{d}$, of stochastic currents corresponding to fractional Brownian motion with Hurst parameter $H\in(0,1)$. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and $d\ge1$ we show that the kernel $ξ(x)$ is well-defined as a Hida distribution for all $H\in(0,1)$. For $x=0$ and $d=1$, $ξ(0)$ is a Hida distribution for all $H\in(0,1)$. For $d\ge2$, then $ξ(0)$ is a Hida distribution only for $H\in(0,1/d)$. For $d=1$, $x \neq 0$, and $H \in (0,1)$, we show that $ξ(x) \in \mathcal{G}'$, the space of regular generalized functions. Elements of the space $\mathcal{G}'$ and elements from the negative Sobolev--Watanabe distribution spaces share the property that partial sums of their chaos decomposition are square integrable functions. More precisely, we show that $ξ(x) \in \mathcal{G}_{-s} \subset \mathcal{G}'$ for $x \neq 0$, $H \in (0,1)$, and all $s > 0$. |
| title | Stochastic Currents of Fractional Brownian Motion: Existence and Regularity |
| topic | Probability 60H40, 60J65, 60G22, 46F25 |
| url | https://arxiv.org/abs/2408.10936 |