Laplace operators in deRham complexes associated with measures on configuration spaces

TY - JOUR

T1 - Laplace operators in deRham complexes associated with measures on configuration spaces

AU - Albeverio, S.

AU - Daletskii, A.

AU - Lytvynov, E.

AU - Kondratiev, Y.

PY - 2002/12

Y1 - 2002/12

N2 - Let GX denote the space of all locally finite configurations in a complete, stochastically complete, connected, oriented Riemannian manifold X, whose volume measure m is infinite. In this paper, we construct and study spaces L2µOn of differential n-forms over GX that are square integrable with respect to a probability measure µ on GX. The measure µ is supposed to satisfy the condition Sm' (generalized Mecke identity) well known in the theory of point processes. On L2µOn, we introduce bilinear forms of Bochner and deRham type. We prove their closabilty and call the generators of the corresponding closures the Bochner and deRham Laplacian, respectively. We prove that both operators contain in their domain the set of all smooth local forms. We show that, under a rather general assumption on the measure µ, the space of all Bochner-harmonic µ-square-integrable forms on GX consists only of the zero form. Finally, a Weitzenböck type formula connecting the Bochner and deRham Laplacians is obtained. As examples, we consider (mixed) Poisson measures, Ruelle type measures on Image , and Gibbs measures in the low activity–high temperature regime, as well as Gibbs measures with a positive interaction potential on GX.

AB - Let GX denote the space of all locally finite configurations in a complete, stochastically complete, connected, oriented Riemannian manifold X, whose volume measure m is infinite. In this paper, we construct and study spaces L2µOn of differential n-forms over GX that are square integrable with respect to a probability measure µ on GX. The measure µ is supposed to satisfy the condition Sm' (generalized Mecke identity) well known in the theory of point processes. On L2µOn, we introduce bilinear forms of Bochner and deRham type. We prove their closabilty and call the generators of the corresponding closures the Bochner and deRham Laplacian, respectively. We prove that both operators contain in their domain the set of all smooth local forms. We show that, under a rather general assumption on the measure µ, the space of all Bochner-harmonic µ-square-integrable forms on GX consists only of the zero form. Finally, a Weitzenböck type formula connecting the Bochner and deRham Laplacians is obtained. As examples, we consider (mixed) Poisson measures, Ruelle type measures on Image , and Gibbs measures in the low activity–high temperature regime, as well as Gibbs measures with a positive interaction potential on GX.

U2 - 10.1016/S0393-0440(02)00221-8

DO - 10.1016/S0393-0440(02)00221-8

M3 - Article

SN - 0393-0440

VL - 47

SP - 259

EP - 302

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

IS - 2-3

ER -