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Laplace operators on the cone of Radon measures

Yuri Kondratiev, Eugene Lytvynov Orcid Logo, Anatoly Vershik

Journal of Functional Analysis, Volume: 269, Issue: 9, Pages: 2947 - 2976

Swansea University Author: Eugene Lytvynov Orcid Logo

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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...

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Published in: Journal of Functional Analysis
Published: 2015
URI: https://cronfa.swan.ac.uk/Record/cronfa23990
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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 SMA 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 COLLEGE CODE SMA Swansea University 2019-05-23T08:20:47.2811960 2015-10-26T17:51:12.0564134 College of 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_***_0000-0001-9685-7727
author Eugene, Lytvynov
author2 Yuri Kondratiev
Eugene Lytvynov
Anatoly Vershik
format Journal article
container_title Journal of Functional Analysis
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container_start_page 2947
publishDate 2015
institution Swansea University
doi_str_mv 10.1016/j.jfa.2015.06.007
college_str College of Science
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hierarchy_top_id collegeofscience
hierarchy_top_title College of Science
hierarchy_parent_id collegeofscience
hierarchy_parent_title College of Science
department_str Mathematics{{{_:::_}}}College of Science{{{_:::_}}}Mathematics
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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 2015-12-31T03:42:45Z
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score 10.854061