Journal article 1543 views
Laplace operators on the cone of Radon measures
Yuri Kondratiev,
Eugene Lytvynov 
,
Anatoly Vershik
,
Anatoly Vershik
Journal of Functional Analysis, Volume: 269, Issue: 9, Pages: 2947 - 2976
Swansea University Author:
Eugene Lytvynov
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DOI (Published version): 10.1016/j.jfa.2015.06.007
Abstract
We consider the infinite-dimensional Lie group $\mathfrak G$ which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold $X$ and the commutative multiplicative group of functions on $X$. The group $\mathfrak G$ naturally acts on the space $\M(X)$ of R...
| Published in: | Journal of Functional Analysis |
|---|---|
| Published: |
2015
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa23990 |
| first_indexed |
2015-10-27T01:55:23Z |
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| last_indexed |
2019-06-05T09:57:54Z |
| id |
cronfa23990 |
| recordtype |
SURis |
| fullrecord |
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| spelling |
2019-05-23T08:20:47.2811960 v2 23990 2015-10-26 Laplace operators on the cone of Radon measures e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2015-10-26 MACS We consider the infinite-dimensional Lie group $\mathfrak G$ which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold $X$ and the commutative multiplicative group of functions on $X$. The group $\mathfrak G$ naturally acts on the space $\M(X)$ of Radon measures on $X$. We would like to define a Laplace operator associated with a natural representation of $\mathfrak G$ in $L^2(\M(X),\mu)$. Here $\mu$ is assumed to be the law of a measure-valued L\'evy process. A unitary representation of the group cannot be determined, since the measure $\mu$ is not quasi-invariant with respect to the action of the group $\mathfrak G$. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group $\mathfrak G$ (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on $\M(X)$ whose generator is the Laplace operator. Journal Article Journal of Functional Analysis 269 9 2947 2976 31 12 2015 2015-12-31 10.1016/j.jfa.2015.06.007 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2019-05-23T08:20:47.2811960 2015-10-26T17:51:12.0564134 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Yuri Kondratiev 1 Eugene Lytvynov 0000-0001-9685-7727 2 Anatoly Vershik 3 |
| title |
Laplace operators on the cone of Radon measures |
| spellingShingle |
Laplace operators on the cone of Radon measures Eugene Lytvynov |
| title_short |
Laplace operators on the cone of Radon measures |
| title_full |
Laplace operators on the cone of Radon measures |
| title_fullStr |
Laplace operators on the cone of Radon measures |
| title_full_unstemmed |
Laplace operators on the cone of Radon measures |
| title_sort |
Laplace operators on the cone of Radon measures |
| author_id_str_mv |
e5b4fef159d90a480b1961cef89a17b7 |
| author_id_fullname_str_mv |
e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov |
| author |
Eugene Lytvynov |
| author2 |
Yuri Kondratiev Eugene Lytvynov Anatoly Vershik |
| format |
Journal article |
| container_title |
Journal of Functional Analysis |
| container_volume |
269 |
| container_issue |
9 |
| container_start_page |
2947 |
| publishDate |
2015 |
| institution |
Swansea University |
| doi_str_mv |
10.1016/j.jfa.2015.06.007 |
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Faculty of Science and Engineering |
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facultyofscienceandengineering |
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Faculty of Science and Engineering |
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facultyofscienceandengineering |
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Faculty of Science and Engineering |
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School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics |
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0 |
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| description |
We consider the infinite-dimensional Lie group $\mathfrak G$ which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold $X$ and the commutative multiplicative group of functions on $X$. The group $\mathfrak G$ naturally acts on the space $\M(X)$ of Radon measures on $X$. We would like to define a Laplace operator associated with a natural representation of $\mathfrak G$ in $L^2(\M(X),\mu)$. Here $\mu$ is assumed to be the law of a measure-valued L\'evy process. A unitary representation of the group cannot be determined, since the measure $\mu$ is not quasi-invariant with respect to the action of the group $\mathfrak G$. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group $\mathfrak G$ (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on $\M(X)$ whose generator is the Laplace operator. |
| published_date |
2015-12-31T18:50:20Z |
| _version_ |
1821523124920778752 |
| score |
11.047674 |