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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!
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 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. College of Science 9 2947 2976