2D- stochastic currents over the Wiener sheet

TL;DR

This paper introduces a new concept of 2D-stochastic currents over the Wiener sheet using stochastic calculus and chaos expansion, analyzing their regularity in both probabilistic and spatial variables.

Contribution

It defines 2D-stochastic currents over the Brownian sheet and studies their regularity properties in Watanabe and Sobolev spaces, bridging stochastic calculus and geometric measure theory.

Findings

Defined 2D-stochastic current over the Brownian sheet

Analyzed regularity in Watanabe spaces

Examined spatial regularity in Sobolev spaces

Abstract

By using stochastic calculus for two-parameter processes and chaos expansion into multiple Wiener-It\^o integrals, we define a 2D-stochastic current over the Brownian sheet. This concept comes from geometric measure theory. We also study the regularity of the stochastic current with respect to the randomness variable in the Watanabe spaces and with respect to the spatial variable in the deterministic Sobolev spaces.