Seminar “Sobolev spaces and spaces of smooth functions on a Hilbert space equipped with a translationally invariant measure”
6 October at 18:00 MSK
To study random walks in a Hilbert space we endowed the last one with a shift invariant measure. According to Weyl's theorem, the Lebesgue measure does not exist in an infinite-dimensional Hilbert space. Thus a measure is considered as a non-negative additive function of a set defined on a ring of subsets of a Hilbert space. The space of functions that are integrable with respect to the translationally invariant measure is introduced and the properties of argument shift operators are studied. It is shown that the result of averaging of shift operators on a random vector with Gaussian distributions is a semigroup of self-adjoint compressions. This averaged semigroup resolves the infinite- dimensional diffusion equation. The criterion of strong continuity of a semigroup is established. Using the introduced diffusion semigroup, we define Sobolev spaces and the space of smooth functions on a Hilbert space. The conditions of embedding and conditions of dense embedding of the space of smooth functions into the Sobolev space are obtained, and examples of violation of the embedding density are given.
Professor V. Zh. Sakbaev ( Moscow Institute of Physics and Technology, Moscow).
Title of the talk: Sobolev spaces and spaces of smooth functions on a Hilbert space equipped with a translationally invariant measure.