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

Abstract

This article extends the celebrated Sewing lemma, known from the theory of rough paths, to multi-parameter fields on hyper-cubes. We use this to construct Young integrals for multi-parameter Holder fields on general domains $$\left[ 0,T\right] ^{k}$$ with $$k\ge 1$$ taking values in $${\mathbb {R}}^{d}$$ . Moreover, we show existence, uniqueness and stability of some particular types of hyperbolic SPDEs driven by space-time Holder noise in a Young regime.