Remarks on CCR and CAR flows over closed convex cones

For a closed convex cone [Formula: see text] in [Formula: see text] which is spanning and pointed, i.e. [Formula: see text] and [Formula: see text] we consider a family of [Formula: see text]-semigroups over [Formula: see text] consisting of a certain family of CCR flows and CAR flows over [Formula: see text] and classify them up to the cocycle conjugacy.

Quadratic-exponential functionals of Gaussian quantum processes

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen–Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

Characters of infinite-dimensional quantum classical groups: BCD cases

We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

On the Bogolyubov endomorphisms of the renormalized square of white noise algebra

We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.

Some strong limit theorems for weighted sums of measurable operators

The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.

A characterization of Riesz-dual sequences which are near-Markushevich bases

The concept of Riesz-duals of a frame is a recently introduced concept with broad implications to frame theory in general, as well as to the special cases of Gabor and wavelet analysis. In this paper, we introduce various alternative Riesz-duals, with a focus on what we call Riesz-duals of type I and II. Next, we provide some characterizations of Riesz-dual sequences in Banach spaces. A basic problem of interest in connection with the study of Riesz-duals in Banach spaces is that of characterizing those Riesz-duals which can essentially be regarded as M-basis. We give some conditions under which an Riesz-dual sequence to be an M-basis for [Formula: see text].

Some noncommutative subsequential weighted individual ergodic theorems

This paper is devoted to studying individual ergodic theorems for subsequential weighted ergodic averages on the noncommutative [Formula: see text]-spaces associated to a semifinite von Neumann algebra [Formula: see text]. In particular, we establish the convergence of these averages along sequences with density one and certain types of block sequences with positive lower density, and we extend known results along uniform sequences in the sense of Brunel and Keane.

Schrödinger dynamics and optimal transport of measures

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

On operator-valued infinitesimal Boolean and monotone independence

We introduce the notion of operator-valued infinitesimal (OVI) Boolean independence and OVI monotone independence. Then we show that OVI Boolean (respectively, monotone) independence is equivalent to the operator-valued (OV) Boolean (respectively, monotone) independence over an algebra of [Formula: see text] upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (respectively, monotone) additive convolution by reducing it to the OV case. We also define OVI Boolean and monotone cumulants and study their basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence.

Sensitivity analysis in the infinite dimensional Heston model

We consider the infinite dimensional Heston stochastic volatility model proposed in Ref. 7. The price of a forward contract on a non-storable commodity is modeled by a generalized Ornstein–Uhlenbeck process in the Filipović space with this volatility. We prove a representation formula for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning of the Greeks. For this we use infinite dimensional Malliavin calculus and a randomization technique.