Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus

AbstractWe give a rigorous version of the functional calculus developed by R. Feynman in relation to his path integral formulation of Quantum Mechanics. Our approach is Euclidean but distinct from the one founded on the Feynman-Kac formula. It uses two basic ingredients: a new probabilistic interpretation of the classical heat equation, introduced recently in the framework of Euclidean Quantum Mechanics, and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation