A $K$-rough path above the space-time fractional Brownian motion

TL;DR

This paper constructs a $K$-rough path over space-time and spatial fractional Brownian motions in any dimension, enabling a rigorous interpretation and unique solution for the renormalized parabolic Anderson model, including spatial fractional noise.

Contribution

It introduces a novel construction of $K$-rough paths for fractional Brownian motions in arbitrary dimensions, facilitating analysis of related stochastic PDEs.

Findings

Successfully constructs $K$-rough paths for fractional Brownian motions.

Provides a unique, renormalized solution to the parabolic Anderson model.

Extends analysis to spatial fractional noise cases.

Abstract

We construct a $K$-rough path above either a space-time or a spatial fractional Brownian motion, in any space dimension $d$. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.