Infinite Dimensional Analysis, Quantum Probability and Related Topics

We study the so-called Nonconventional Ergodic Theorem for noncommutative generic measures introduced by Furstenberg in classical ergodic theory, and the relative three-point multiple correlations of arbitrary length arising from several situations of interest in quantum case. We deal with the diagonal state canonically associated to the product state (i.e. quantum "diagonal measures") in the ergodic situation, and with the case concerning convex combinations (i.e. direct integral) of diagonal measures in nonergodic one. We also treat in the full generality the case of compact dynamical systems, that is when the unitary generating the dynamics in the Gelfand–Naimark–Segal representation is almost periodic. In all the above-mentioned situations, we provide the explicit formula for the involved ergodic average. Such an explicit knowledge of the limit of the three-point correlations is naturally relevant for the investigation of the long time behavior of a dynamical system.

In this paper we prove the existence of martingale solutions for the 2D stochastic fractional vorticity Navier–Stokes equation driven by space-time white noise for α ∈ (½, 1] and the 2D stochastic quasi-geostrophic equation on 𝕋² for α ∈ (0, 1] driven by non-trace class noise. We also cover the case driven by non-trace class multiplicative noise for all α ∈ (0, 1].

This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space of Potthoff–Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.

The quantum symmetry group of the inductive limit of C*-algebras equipped with orthogonal filtrations is shown to be the projective limit of the quantum symmetry groups of the C*-algebras appearing in the sequence. Some explicit examples of such projective limits are studied, including the case of quantum symmetry groups of the duals of finite symmetric groups, which do not fit directly into the framework of the main theorem and require further specific study. The investigations reveal a deep connection between quantum symmetry groups of discrete group duals and the doubling construction for Hopf algebras.

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ–g-frame, where Θ is a bounded operator on a Hilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ–g-frames by considering Θ–g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

We prove that the binomial sequence is positive definite if and only if either p ≥ 1, -1 ≤ r ≤ p - 1 or p ≤ 0, p - 1 ≤ r ≤ 0 and that the Raney sequence is positive definite if and only if either p ≥ 1, 0 ≤ r ≤ p or p ≤ 0, p - 1 ≤ r ≤ 0 or else r = 0. The corresponding probability measures are denoted by ν(p, r) and μ(p, r) respectively. We prove that if p > 1 is rational and -1 < r ≤ p - 1 then the measure ν(p, r) is absolutely continuous and its density V_p,r(x) can be represented as Meijer G-function. In some cases V_p,r is an elementary function. We show that for p > 1 the measures ν(p,-1) and ν(p,0) are certain free convolution powers of the Bernoulli distribution.