Construction and Skorohod representation of a fractional K-rough path

TL;DR

This paper constructs a fractional K-rough path for a heat equation with fractional noise, extending roughness domain coverage and providing a Skorohod integral representation, advancing stochastic analysis techniques.

Contribution

It introduces the concept of a K-rough path bridging regularity structures and rough paths theory, enabling analysis of fractional noise in heat equations.

Findings

Constructed a K-rough path at order three covering the full roughness domain.

Provided a Skorohod integral representation of the K-rough path.

Extended the analysis to space-time fractional noise scenarios.

Abstract

We go ahead with the study initiated in [3] about a heat-equation model with non-linear perturbation driven by a space-time fractional noise. Using general results from Hairer's theory of regularity structures, the analysis reduces to the construction of a so-called K-rough path (above the noise), a notion we introduce here as a compromise between regularity structures formalism and rough paths theory. The exhibition of such a K-rough path at order three allows us to cover the whole roughness domain that extends up to the standard space-time white noise situation. We also provide a representation of this abstract K-rough path in terms of Skorohod stochastic integrals.