Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks
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 b...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2018-06-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/shift-invariant-measures-on-infinite-dimensional-403659.html |
| _version_ | 1826904236559958016 |
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| author | V.Zh. Sakbaev D.V. Zavadsky |
| author_facet | V.Zh. Sakbaev D.V. Zavadsky |
| author_sort | V.Zh. Sakbaev |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T06:48:23Z |
| format | Article |
| id | doaj.art-c749357c61f84d3483e96d67b6d3e317 |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2025-02-17T08:08:13Z |
| publishDate | 2018-06-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj.art-c749357c61f84d3483e96d67b6d3e3172025-01-03T00:06:23ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982018-06-011602384391Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walksV.Zh. Sakbaev0D.V. Zavadsky1Moscow Institute of Physics and Technology, Dolgoprudny, 141701 RussiaMoscow Institute of Physics and Technology, Dolgoprudny, 141701 RussiaAveraging 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.https://kpfu.ru/shift-invariant-measures-on-infinite-dimensional-403659.htmlstrongly continuous semigroupsaveraging of operator semigroupsshift-invariant measuressquare integrable functions |
| spellingShingle | V.Zh. Sakbaev D.V. Zavadsky Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks Учёные записки Казанского университета: Серия Физико-математические науки strongly continuous semigroups averaging of operator semigroups shift-invariant measures square integrable functions |
| title | Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks |
| title_full | Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks |
| title_fullStr | Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks |
| title_full_unstemmed | Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks |
| title_short | Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks |
| title_sort | shift invariant measures on infinite dimensional spaces integrable functions and random walks |
| topic | strongly continuous semigroups averaging of operator semigroups shift-invariant measures square integrable functions |
| url | https://kpfu.ru/shift-invariant-measures-on-infinite-dimensional-403659.html |
| work_keys_str_mv | AT vzhsakbaev shiftinvariantmeasuresoninfinitedimensionalspacesintegrablefunctionsandrandomwalks AT dvzavadsky shiftinvariantmeasuresoninfinitedimensionalspacesintegrablefunctionsandrandomwalks |