Stochastic Currents of Fractional Brownian Motion: Existence and Regularity

By using white noise analysis, we study the integral kernel $\xi (x)$, $x\in {\mathbb{R}}^d$, of stochastic currents corresponding to fractional Brownian motion with Hurst parameter $H\in (0,1)$. For $x\in {\mathbb{R}}^d\mathrm{\backslash }\{0\}$ and $d\ge 1$ we show that the kernel $\xi (x)$ is well-defined as a Hida distribution for all $H\in (0,1)$. For $x=0$ and $d=1$, $\xi (0)$ is a Hida distribution for all $H\in (0,1)$. For $d\ge 2$, then $\xi (0)$ is a Hida distribution only for $H\in (0,1\mathrm{/}d)$. For $d=1$, $x\mathrm{\ne }0$, and $H\in (0,1)$, we show that $\xi (x)\in {\mathcal{G}}^{\mathrm{\prime }}$, the space of regular generalized functions. Elements of the space ${\mathcal{G}}^{\mathrm{\prime }}$ 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 $\xi (x)\in {\mathcal{G}}_{-s}\subset {\mathcal{G}}^{\mathrm{\prime }}$ for $x\mathrm{\ne }0$, $H\in (0,1)$, and all $s>0$.