Infinite Dimensional Analysis, Quantum Probability and Related Topics

We consider a family of infinite dimensional Laplace operators which contains the classical Lévy–Laplacian. We prove a representation of these operators as a quadratic functions of quantum stochastic processes. Particularly, for the classical Lévy–Laplacian, the following formula is proved: Δ_L = lim_ε→0 ∫_‖s-t‖<ε b_sb_tdsdt, where b_t is the annihilation process.

The authors have previously studied multiplicative renormalization method (MRM) for generating functions of orthogonal polynomials. In particular, they have determined all MRM-applicable measures for renormalizing functions h(x) = e^x, h(x) = (1 - x)^-κ, κ = 1/2, 1, 2. For the cases h(x) = e^x and (1 - x)^-1, there are very large classes of MRM-applicable measures. For the other two cases κ = 1/2, 2, MRM-applicable measures belong to special classes of a certain kind of beta distributions. In this paper, we determine all MRM-applicable measures for h(x) = (1 - x)^-κ with κ ≠ 0, 1, 1/2.

We propose the path-integral technique to derive the characteristic function of the limiting distribution of the unit root test in a first order autoregressive model. Our results provide a new and useful approach to obtain the closed form of the characteristic function of a random variable associated with the limiting distribution, which is realized as a ratio of Brownian functionals on the classical Wiener space.

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree-like behavior and is locally constant for some time periods. It is natural to apply this kind of models for the investigation of avalanche processes and punctuated equilibrium as well as fractal-like analysis of time series generated by measurement of pressure in oil wells.

Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by Kretschmann, Schlingemann and Werner. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials.

Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on ℝ^d\{0}.

An open C*-dynamical systems is a triple {𝔄, Φ, φ} where 𝔄 is a C* algebra with unit, φ as its state, Φ : 𝔄 → 𝔄 a unital completely positive map with φ ◦ Φ = φ. Such system is called purely implementable if there exists a representation π of 𝔄 in the bounded operators on Hilbert spaces , an isometry and a V-invariant vector such that π(Φ(a)) = V*π(a)V and φ(a) = 〈Ω, π(a)Ω〉 for all a ∈ 𝔄, with Ω cyclic vector for the algebra generated by π(𝔄) and V. The quadruple is said be a covariant GNS representation of the open C*-dynamical system. We prove that every open C*-dynamical system is purely implementable. In the case when the dynamics Φ is a *-homomorphism this result was obtained by Niculescu Ströh and Zsidó. Moreover we prove that, in the case of *-homomorphisms, the above-mentioned construction provides an alternative construction of a minimal dilation in the sense of Kümmerer in which the dilation of the dynamics Φ has the same ergodic properties as Φ.

We prove a disintegration theorem for the Gaussian Radon transform Gf on Banach spaces and use the Segal–Bargmann transform on abstract Wiener spaces to find a procedure to obtain f from its Gaussian Radon transform Gf.