A White Noise Approach to Stochastic Currents of Brownian Motion

TL;DR

This paper investigates the properties of stochastic currents of Brownian motion using white noise analysis, establishing conditions under which these currents are Hida distributions and highlighting differences in behavior at the origin.

Contribution

The paper introduces a white noise analysis framework to study stochastic currents of Brownian motion and characterizes when these currents are Hida distributions.

Findings

Stochastic currents are Hida distributions for non-zero points.

At the origin in higher dimensions, stochastic currents are not Hida distributions.

The analysis provides new insights into the structure of Brownian motion currents.

Abstract

In this paper we study stochastic currents of Brownian motion $\xi(x)$, $x\in\mathbb{R}^{d}$, by using white noise analysis. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and for $x=0\in\mathbb{R}$ we prove that the stochastic current $\xi(x)$ is a Hida distribution. Moreover for $x=0\in\mathbb{R}^{d}$ with $d>1$ we show that the stochastic current is not a Hida distribution.