Gradient-type noises I - partial and hybrid integrals

The final objective of our study is to propose a finite-dimensional approach to systems of parabolic partial differential equations perturbed by low-order noises of Brownian or fractional Brownian type. The present article is the preparatory first part, where we introduce partial pathwise integrals over D and (a, b) x D, where D is a smooth bounded domain in R-n. Corresponding stochastic versions appear as limit cases.