Second quantisation for skew convolution products of infinitely divisible measures

Suppose λ1 and λ2 are infinitely divisible Radon measures on real Banach spaces E1 and E2, respectively and let T : E1 → E2 be a Borel measurable mapping so that T(λ1) * ρ = λ2 for some Radon probability measure ρ on E2. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the "transition operator" PT : Lp(E2, λ2) → Lp(E1, λ1) given by [Formula: see text] as the second quantisation of a contraction operator acting between suitably chosen "reproducing kernel Hilbert spaces" associated with λ1 and λ2.

Mackey Continuity of Characteristic Functionals

Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional.

The Topological Support of Gauss Measure on Hilbert Space

Let X be a Hilbert space. The topological support of a Radon probability measure P on X is the least closed subset M of X that carries the total measure 1.