Laplace operators on the cone of Radon measures

We consider the infinite-dimensional Lie group GG 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 GG naturally acts on the space M(X)M(X) of Radon measures on X  . We would like to define a Laplace operator associated with a natural representation of GG in L2(M(X),μ)L2(M(X),μ). Here μ is assumed to be the law of a measure-valued Lévy process. A unitary representation of the group cannot be determined, since the measure μ   is not quasi-invariant with respect to the action of the group GG. 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 GG (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)M(X) whose generator is the Laplace operator.