scholarly journals De Rham–Hodge decomposition and vanishing of harmonic forms by derivation operators on the Poisson space

2016 ◽  
Vol 19 (02) ◽  
pp. 1650010 ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Poisson Process ◽  
Covariant Derivative ◽  
Probability Space ◽  
Differential Forms ◽  
Hodge Decomposition ◽  
Wiener Space ◽  
Harmonic Forms ◽  
Space Setting ◽  
De Rham ◽  
Poisson Space

We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham–Hodge–Kodaira decomposition as well as Weitzenböck and Clark–Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms.

scholarly journals ITÔ–WIENER CHAOS AND THE HODGE DECOMPOSITION ON AN ABSTRACT WIENER SPACE

2013 ◽  
Vol 16 (01) ◽  
pp. 1350008
Author(s):  
YUXIN YANG
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Fock Space ◽  
Hodge Decomposition ◽  
Wiener Space ◽  
Space Representation ◽  
Abstract Wiener Space ◽  
Wiener Chaos ◽  
De Rham ◽  
Symplectic Connections

Using the structure of the Boson-Fermion Fock space and an argument taken from [P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley and L. Schwachhofer, Symplectic connections, Int. J. Geom. Meth. Mod. Phys.3 (2006) 375–420], we give a new proof of the triviality of the L2 cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [De Rham–Hodge–Kodaira's decomposition on an abstract Wiener space, J. Math. Kyoto. Univ.26(2) (1986) 191–202]. We apply the representation theory of the symmetric group to characterize the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.

DE RHAM WAVE EQUATION FOR TENSOR VALUED p-FORMS

2003 ◽  
Vol 12 (08) ◽  
pp. 1363-1384 ◽  
Author(s):  
DONATO BINI ◽  
CHRISTIAN CHERUBINI ◽  
ROBERT T. JANTZEN ◽  
REMO RUFFINI
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Wave Equation ◽  
Covariant Derivative ◽  
Differential Forms ◽  
Black Hole Spacetimes ◽  
Metric Connection ◽  
De Rham ◽  
Integral Spin ◽  
Curvature Tensors

The de Rham Laplacian Δ (dR) for differential forms is a geometric generalization of the usual covariant Laplacian Δ, and it may be extended naturally to tensor-valued p-forms using the exterior covariant derivative associated with a metric connection. Using it the wave equation satisfied by the curvature tensors in general relativity takes its most compact form. This wave equation leads to the Teukolsky equations describing integral spin perturbations of black hole spacetimes.

scholarly journals The Bezout-Corona Problem Revisited: Wiener Space Setting

2015 ◽  
Vol 10 (1) ◽  
pp. 115-139
Author(s):  
G. J. Groenewald ◽  
S. ter Horst ◽  
M. A. Kaashoek
Keyword(s):  
Wiener Space ◽  
Corona Problem ◽  
Space Setting

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

scholarly journals The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

scholarly journals The Double Complex of a Blow-up

2019 ◽  
Author(s):  
Jonas Stelzig
Keyword(s):  
Blow Up ◽  
Differential Forms ◽  
Complex Manifolds ◽  
Double Complex ◽  
Smooth Complex ◽  
Projective Bundles ◽  
De Rham ◽  
Suitable Notion ◽  
Complex Valued ◽  
Compact Complex Manifolds

AbstractWe compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for “all” cohomologies naturally associated with this complex (in particular, de Rham, Dolbeault, Bott–Chern, and Aeppli).

VANISHING THEOREM OF THE HODGE–KODAIRA OPERATOR FOR DIFFERENTIAL FORMS ON A CONVEX DOMAIN OF THE WIENER SPACE

2003 ◽  
Vol 06 (supp01) ◽  
pp. 53-63 ◽  
Author(s):  
ICHIRO SHIGEKAWA
Keyword(s):  
Boundary Condition ◽  
Convex Domain ◽  
Bilinear Form ◽  
Differential Forms ◽  
Wiener Space ◽  
Harmonic Form ◽  
Vanishing Theorem ◽  
The Absolute

We discuss the vanishing theorem on a convex domain of the Wiener space. We show that there is no harmonic form satisfying the absolute boundary condition. Our method relies on an expression of the bilinear form associated with the Hodge–Kodaira operator.

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