Differentiable measures and the Malliavin calculus

Abstract

IntroductionAmong the most notable events in the nonlinear functional analysis in the past two decades, one can mention the development of the theory of differentiable measures and the creation of the Malliavin calculus. These two theories can be regarded as infinite-dimensional analogs of such classical fields as geometric measure theory, the theory of Sobolev spaces, and the theory of generalized functions.The theory of differentiable measures was suggested by S. V. Fomin in his report at the International Congress of Mathematicians in Moscow in 1966 as an infinite-dimensional substitute for the Sobolev-Schwartz theory of distributions (see [213-215, 36, 37]). Fomin realized that it is natural to consider four spaces, namely, a certain space of functions S, its dual S', a certain space of measures M, and its dual M' in infinite dimensions instead of a pair of spaces (test functions-generalized functions). In the finite-dimensional case, M is identified with S by means of the Lebesgue measure, which allows us to represent measures via their densities. It is the lack of any analogs of the Lebesgue measure that destroys this identification in infinite dimensions. The Fourier transform then acts between S and M and between M' and S'. Thus, a theory of pseudodifferential operators can be developed: the initial Fomin objective was to study infinite-dimensional partial differential equations. However, as often happens with fruitful ideas, the theory of differentiable measures overgrew the initial framework. This theory became an efficient tool in a wide variety of the most diverse applications such as stochastic analysis, quantum field theory, and nonlinear analysis. It is a rapidly developing field which is rich in challenging problems of great importance that provide for better understanding of the nature of infinite-dimensional phenomena. It is worth mentioning that before the pioneering Fomin works similar ideas appeared in several papers by T. Pitcher on the distributions of random processes in functional spaces. Pitcher investigated even more general situations, namely, he studied the differentiability of a family of measures /h = /~ o Tf I generated by a family of transformations Tt of a fixed measure #. Fomin's differentiability corresponds to the case where the measures #t are the shifts #th of the measure # in a linear space. However, the theory constructed for this special case is much richer.In the middle of the seventies, Malliavin [367] suggested an elegant method of proving the smoothness of the transition probabilities of finite-dimensional diffusions. The essence of the method was to consider the transition probabilities Pt as images of the Wiener measure under some nonlinear transformations (generated by stochastic differential equations) and then to apply an integration-by-parts formula on the Wiener space leading to estimates of the generalized derivatives of Pt ensuring the membership in C~ A probabilistic proof of HSrmander's theorem on hypoelliptic second-order operators was given as an applicat...

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