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Laplace operators on the cone of Radon measures

Yuri Kondratiev, Eugene Lytvynov , Anatoly Vershik

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

Swansea University Author:

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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 2015 https://cronfa.swan.ac.uk/Record/cronfa23990 No Tags, Be the first to tag this record!
first_indexed 2015-10-27T01:55:23Z 2019-06-05T09:57:54Z cronfa23990 SURis 2019-05-23T08:20:47.2811960v2239902015-10-26Laplace operators on the cone of Radon measurese5b4fef159d90a480b1961cef89a17b70000-0001-9685-7727EugeneLytvynovEugene Lytvynovtruefalse2015-10-26SMAWe 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 ArticleJournal of Functional Analysis269929472976311220152015-12-3110.1016/j.jfa.2015.06.007COLLEGE NANMEMathematicsCOLLEGE CODESMASwansea University2019-05-23T08:20:47.28119602015-10-26T17:51:12.0564134College of ScienceMathematicsYuriKondratiev1EugeneLytvynov0000-0001-9685-77272AnatolyVershik3 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 Laplace operators on the cone of Radon measures Laplace operators on the cone of Radon measures Eugene, Lytvynov Laplace operators on the cone of Radon measures Laplace operators on the cone of Radon measures Laplace operators on the cone of Radon measures Laplace operators on the cone of Radon measures Laplace operators on the cone of Radon measures e5b4fef159d90a480b1961cef89a17b7 e5b4fef159d90a480b1961cef89a17b7_***_Eugene, Lytvynov_***_0000-0001-9685-7727 Eugene, Lytvynov Yuri Kondratiev Eugene Lytvynov Anatoly Vershik Journal article Journal of Functional Analysis 269 9 2947 2015 Swansea University 10.1016/j.jfa.2015.06.007 College of Science collegeofscience College of Science collegeofscience College of Science Mathematics{{{_:::_}}}College of Science{{{_:::_}}}Mathematics 0 0 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. 2015-12-31T03:42:45Z 1723077718775431168 10.854061