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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/2408.10936 |
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Table of 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 $s > 0$.