Brownian and fractional Brownian stochastic currents via Malliavin calculus

TL;DR

This paper investigates the existence and regularity of stochastic currents driven by Brownian and fractional Brownian motions using Malliavin calculus, including multidimensional cases and regularity comparisons.

Contribution

It introduces a framework for analyzing stochastic currents via Malliavin calculus, extending to multidimensional and multiparameter cases, and compares their regularity in different functional spaces.

Findings

Established existence of stochastic currents as Skorohod integrals.

Analyzed regularity in negative Sobolev and Watanabe spaces.

Extended results to multidimensional and multiparameter settings.

Abstract

By using Malliavin calculus and multiple Wiener-It\^o integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.