Stochastic Currents of Fractional Brownian Motion

TL;DR

This paper investigates the integral kernel of stochastic currents associated with fractional Brownian motion using white noise analysis, establishing conditions under which the kernel is a Hida distribution across different dimensions and Hurst parameters.

Contribution

It provides new conditions for the well-definedness of the integral kernel as a Hida distribution for fractional Brownian motion in various dimensions and Hurst parameters.

Findings

Kernel is a Hida distribution for all H in (0,1/2] when x ≠ 0.

For d=1, kernel is a Hida distribution at x=0 for all H in (0,1).

For d≥2, kernel at x=0 is a Hida distribution only for H in (0,1/d).

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 $2N(H-1)+Hd>1$.