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We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an Itô formula for a diffusion in a distributional setting, and apply it to investigate differentiability-index s and integrability-index p 2 for which the Bochner integral belongs to D s p .doi:10.30757/alea.v15-27 fatcat:xs6if26btveclonfe2x7z7e4du