Laplace operators on the cone of Radon measures

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 $\mathbb M(X)$ of Radon measures on $X$. We would like to define a Laplace operator associated with a natural representation of $\mathfrak G$ in $L^2(\mathbb M(X),\mu)$. 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 $\mathbb M(X)$ whose generator is the Laplace operator.