Stochastic Currents of Fractional Brownian Motion (2408.10936v1)

Abstract: 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 $x\in\mathbb{R}^{d}\backslash{0}$ and $d\ge1$ we show that the kernel $\xi(x)$ is well-defined as a Hida distribution for all $H\in(0,1/2]$. For $x=0$ and $d=1$, $\xi(0)$ is a Hida distribution for all $H\in(0,1)$. For $d\ge2$, then $\xi(0)$ is a Hida distribution only for $H\in(0,1/d)$. To cover the case $H\in[1/d,1)$ we have to truncate the delta function so that $\xi^{(N)}(0)$ is a Hida distribution whenever $2N(H-1)+Hd>1$.