An extension of the sewing lemma to hyper-cubes and hyperbolic equations driven by multi parameter Young fields

TL;DR

This paper extends the sewing lemma to multi-parameter fields on hypercubes, enabling the construction of Young integrals for such fields and establishing results on hyperbolic SPDEs driven by space-time H"older noise.

Contribution

It introduces an extension of the sewing lemma to multi-parameter functions and hyperbolic equations driven by multi-parameter Young fields, filling a gap in stochastic analysis.

Findings

Extended sewing lemma to multi-parameter fields

Constructed Young integrals for multi-parameter H"older fields

Proved existence, uniqueness, and stability of hyperbolic SPDEs in a Young regime

Abstract

This article extends the celebrated sewing lemma to multi-parameter fields on hyper cubes. We use this to construct Young integrals for multi-parameter H\"older fields on general domains [0,1]^{k} with k>=1. Moreover, we show existence, uniqueness and stability of some particular types of hyperbolic SPDE's driven by space-time H\"older noise in a Young regime This article replaces the article "Rough Integration for Fields - with applications to stochastic hyperbolic PDE's". Due to errors in the first article in the section on rough fields, we have decided to divide this article into two parts. The first, is dealing with extending the sewing lemma to general multi parameter functions as well as hyperbolc equations driven by multiparameter fields in a Young regime. The next article will come later and study the rough path type framework implied by the extended sewing lemma.