Malliavin calculus

"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 Hörmanders 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 Hörmanders theorem than is available in the PDE literature ... However Malliavins 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."

"We present an approach that allows one to introduce a Malliavin type calculus for functionals of general Lévy processes and to obtain sufficient conditions for the absolute continuity of solutions of stochastic differential equations with jumps (we do not pose any assumptions about regularity of the intensity of the jumps). Our investigations are motivated by a pioneering idea due to Bismut ... and developed further by many authors. The idea is to extend the Malliavin approach to regularity of Wiener functionals to more general probability spaces by introducing a smooth structure in these spaces in terms of a “differentiation rule”, integration-by-parts formula, and by further applications of the stochastic calculus of variations to smooth functionals with nondegenerate derivatives."

"The mathematical theory now known as Malliavin calculus was first introduced by Paul Malliavin .. as an infinite-dimensional integration by parts technique. The purpose of this calculus was to prove the results about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motiion."

"Malliavin’s work inspired many new results in stochastic analysis. Examples include filtering theorems (Michel ...), a deeper understanding of the Skorohod integral and the development of an anticipating stochastic calculus (Nualart and Paradox ...), an extension of Clark’s formula (Ocone ...), Bismut’s probabilistic analysis of the small-time asymptotics of the heat kernel of the Dirac operator on a Riemannian manifold ... and his subsequent proof of the associated index theorem ..., and a sharp hypoellipticity theorem for Hörmander operators with hypersurfaces of infinite type (Bell and Mohammed ...)."