Analysis and Geometry on Configuration Spaces

A Characterization of Hida Distributions

We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.

Harmonic analysis on configuration space i: general theory

We study the spaces of Poisson, compound Poisson and Gamma noises as special cases of a general approach to non-Gaussian white noise calculus, see Ref. 18. We use a known unitary isomorphism between Poisson and compound Poisson spaces in order to transport analytic structures from Poisson space to compound Poisson space. Finally we study a Fock type structure of chaos decomposition on Gamma space.

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Analysis on Poisson and Gamma spaces

/pdf/an-analytic-characterization-of-symbols-of-operators-on-6i7w98xcfv.pdf

An analytic characterization of symbols of operators on white noise functionals

Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc ([8], [9]), analogously to the works of T Hida ([3], [4], [5]) Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf [10], [11], [12], [13]) Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc in a way completely parallel with the Gaussian case The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case, as will be stated in Section 5 Conversely, those formulae characterize the types of white noises

/pdf/calculus-on-gaussian-and-poisson-white-noises-zs1unfn4d6.pdf

Calculus on Gaussian and Poisson white noises

The first author and S. Takenaka introduced the structure of a Gel’fand triplet ℋ ⊂ (L2 ) ⊂ ℋ* into Hida’s calculus on generalized Brownian functionals [4-7]. They showed that the space ℋ of testing random variables has nice properties. For example, ℋ is closed under multiplication of two elements in ℋ, each element of ℋ is a continuous functional on the basic space ℰ*, in addition it can be considered as an analytic functional, and moreover exp [tΔv] (Δv is Volterra’s Laplacian) is real analytic in t ∊ R as a one-parameter group of operators on ℋ, etc.

/pdf/a-remark-on-the-space-of-testing-random-variables-in-the-17tjq46ze5.pdf

Izumi Kubo

Papers

Calculus on Gaussian and Poisson white noises

A remark on the space of testing random variables in the white noise calculus