In the Hida-Streit approach of functional integral, path integrals are defined as distribution. This allowed to Khandrekhar-Streit to define the Feynman path integral on $R^d$ as a distribution. We choose others functional spaces in order to define path integrals. We study several cases:
-)The case of Feynman path on a manifold.
-)The case of the Bismut-Chern character and heuristic formulas of Atiyah-Witten.
-)The case of the Index theorem for family and its relation with the filtering equation pionneered by Bismut.