Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

Abstract

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

A White Noise Approach to Occupation Times of Brownian Motion

“…For example, it can be used to study local times, self-intersection local times, stochastic current and Feynman integrals, see e.g. [2,3,6,14,15,18]. The derivatives of Donsker's delta distribution has been also studied in [16].…”