کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4589951 | 1334922 | 2015 | 30 صفحه PDF | دانلود رایگان |
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.
Journal: Journal of Functional Analysis - Volume 269, Issue 9, 1 November 2015, Pages 2947–2976