Infinite Dimensional Analysis, Quantum Probability and Related Topics

We prove in this paper that the vacuum expectation of any product of field operator of an interacting Fock space with the interactions λ_n's is driven by pair partitions if and only if each interaction λ_n is a linear combination of permutation operators.

We consider quantum systems that have as their configuration spaces finite-dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to a measure on the Skorokhod space of paths, D [0, ∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman–Kac formula in the local field setting as a consequence of the Hille–Yosida theory of semigroups.

We prove the existence of stationary random fields with linear regressions for q > 1 and thus close an open question posed by W. Bryc et al. We prove this result by describing a discrete one-dimensional conditional distribution and then checking Chapman–Kolmogorov equation. Support of this distribution consists of zeros of certain Al-Salam–Chihara polynomials. To find them we refer to and expose known result concerning addition of q-exponential function. This leads to generalization of a well-known formula , where H_k(x) denotes kth Hermite polynomial.

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on leaving a maximal commutative subalgebra invariant and show that there exist Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for any d > 2.

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we consider an extended stochastic integral and construct elements of a Wick calculus on parametrized Kondratiev-type spaces of generalized functions; consider the interconnection between the extended stochastic integration and the Wick calculus; and give an example of a stochastic equation with a Wick-type nonlinearity. The main results consist of studying the properties of the extended (Skorohod) stichastic integral subject to the particular spaces under consideration; and of studying the properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces. These results are necessary, in particular, in order to describe properties of solutions of normally ordered white noise equations in the "Meixner analysis".

In 2000 Carnovale and Koornwinder defined a q-convolution and proved that for some classes of measures it is associative and commutative. We investigate its positivity preserving properties. One of them is the notion of q-positivity related to q-moments. In this paper we describe an algebraic interpretation of q-positivity which leads us to the definition of (p, q)-convolution. It has a form similar to the q-convolution of Carnovale and Koornwinder coming from a braided algebra. For the new convolution we find an appropriate analogue of Fourier transform and also present a central limit theorem.

In this paper, we apply the completion of squares method to study the optimal investment problem under mean-variance criteria for an insurer. The insurer's risk process is modelled by a classical risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained. Moreover, when H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

We consider the linear stochastic Cauchy problem

Quenched local large deviation is derived for the super-Brownian motion with super-Brownian immigration, in dimension d ≥ 4. At the critical dimension d = 4, the quenched and annealed LDP are of the same speed but are different rate.

We investigate the validity of quantum De Finetti's Theorem for algebraic random processes satisfying the singleton condition.

We first prove a conditional version of the theorem, then we give some natural conditions which guarantee the full validity of De Finetti's Theorem.