Smooth measures on manifolds with a Riemannian structure

The article is devoted to the current and actively developing direction of development of the analysis of smooth measures on smooth infinite-dimensional manifolds. The importance of this direction is dictated by the vast area of its applications, which include infinite-dimensional analysis and a number of sections of mathematical physics of systems with an infinite number of degrees of freedom. In infinite-dimensional analysis, due to the absence of a standard measure such as Lebesgue measure, the spaces of measures play the same role as the spaces of functions. Therefore, in each of these spaces, a differential calculus must be constructed independently. Moreover, the dual object to smooth measures is generalized functions, and to smooth functions — generalized measures. The differential calculus of measures in linear spaces was developed in the works of S. V. Fomin, A. V. Averbukh, O. G. Smolyanov, as well as in the works of A. V. Skorokhod. But in all these works, differentiation of measures in constant directions was considered. Therefore, it is necessary to consider the concept of a derivative measure along a vector field, which makes sense both in linear space and on a smooth manifold. We establish the invariance of the derivative measure along the vector field with respect to smooth transformations and the stability of the logarithmic derivative of the measure with respect to smooth invertible mappings. Moreover, the derivative of the measure along the finite set of vector fields turns out to be symmetric with respect to the vector fields. The corresponding constructions lead to a generalization to non-Gaussian cases of integration formulas by parts, which underlie the construction of extended stochastic integrals. Covariant differential operations on manifolds with a Hilbert–Schmidt structure are considered. The model of such a manifold is the Banach rigging of a real separable Hilbert space. Partial integration formulas for such varieties are closely related to the differential-geometric characteristics of the manifold; they contain the Ricci tensor of the manifold.