Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given. 2000 AMS Mathematics Subject Classification. 60G57, 58A10Contents

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation is given. 1

We investigate nite-time blow-up and stability of semilinear partial dierential equations of the form @w t =@t = w t +t t , w 0 (x) = '(x) 0, x 2 R+ , where is the generator of the standard gamma process and > 0, 2 R, > 0 are constants. We show that any initial value satisfying c 1 x '(x), x > x 0 for some positive constants x 0 ; c 1 ; a 1 , yields a non-global solution if a 1 < 1 + , or if a 1 = 1 + and > 1. If '(x) c 2 x , x > x 0 ; where x 0 ; c 2 ; a 2 > 0, and a 2 > 1 + , then the solution w t is global and satis es 0 w t (x) Ct , x 0, for some constant C > 0. This extends the results previously obtained in the case of -stable generators. Systems of semilinear PDE's with gamma generators are also considered.

The space ΓX of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of square-integrable differential forms over ΓX, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the L 2-cohomology of the underlying manifold X.

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