Distributional It\^o's Formula and Regularization of Generalized Wiener Functionals

We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an It\^o formula for a diffusion in a distributional setting, and apply to investigate differentiability-index $s$ and integrability-index $p \geqslant 2$ for which the Bochner integral belongs to $\mathbb{D}_{p}^{s}$.