Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks
V.Zh. Sakbaev, D.V. Zavadsky
Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia
Abstract
Averaging of random shift operators on a space of the square integrable by shift-invariant measure complex-valued functions on linear topological spaces has been studied. The case of the l∞ space has been considered as an example.
A shift-invariant measure on the l∞ space, which was constructed by Caratheodory's scheme, is σ-additive, but not σ-finite. Furthermore, various approximations of measurable sets have been investigated. One-parameter groups of shifts along constant vector fields in the l∞ space and semigroups of shifts to a random vector, the distribution of which is given by a collection of the Gaussian measures, have been discussed. A criterion of strong continuity for a semigroup of shifts along a constant vector field has been established.
Conditions for collection of the Gaussian measures, which guarantee the semigroup property and strong continuity of averaged one-parameter collection of linear operators, have been defined.
Keywords: strongly continuous semigroups, averaging of operator semigroups, shift-invariant measures, square integrable functions
Acknowledgements. The work was performed according to the Russian Government Program of Competitive Growth of Moscow Institute of Physics and Technology (project 5-100).
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Recieved
October 17, 2017
For citation: Sakbaev V.Zh., Zavadsky D.V. Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 384–391.
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