"The Malliavin calculus refers to a part of Probability theory which can loosely be described as a type of calculus of variations for Brownian motion. It is intimately concerned with the interplay between Markov processes with continuous paths (i.e., diffusions) and partial differential equations. ... What Malliavin did was to provide a probabilistic proof of Hörmander's theorem by constructing a kind of calculus of variations for Brownian motion. This in turn gave probabilistic proofs of the smoothness of the transition densities. This has the advantage of giving probabilistic insight and intuition into what is seen as a fundamental probabilistic result; it has the disadvantage of giving a longer and perhaps harder proof of Hörmander's theorem than is available in the PDE literature ... However Malliavin's methods (credit should also be given to those whose work he built upon such as Gross, Kree, Kuo, Eels, Elworthy, .,. ) are profound, and they are already having ramifications in other areas of probability."