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We translate into the language of semi-group theory Bismut's Calculus on boundary processes (Bismut (1983), Leandre (1989)) which gives regularity result on the heat kernel associated with fractional powers of degenerated Laplacian. We translate into the language of semi-group theory the marriage of Bismut (1983) between the Malliavin Calculus of Bismut type on the underlying diffusion process and the Malliavin Calculus of Bismut type on the subordinator which is a jump process.

Malliavin Calculus of Bismut Type for Fractional Powers of Laplacians in Semi-Group Theory