Infinite Dimensional Analysis, Quantum Probability and Related Topics

An MRM-triple (h(x), ρ(t), B(t)) gives a generating function B(t)h(ρ(t)x) of some orthogonal polynomials on ℝ. In particular, B(t)h(tx) is called the Brenke type.¹⁹ In this paper, we shall determine all MRM-triples and associated Jacobi-Szegö parameters of this type with showing very careful computations in detail. (h(x), t, B(t)) is classified into four categories. In any case, h(x) and B(t) can be expressed in terms of two kinds of q-hypergeometric series, old basic and basic hypergeometric series, _rΦ_s and _rϕ_s, respectively. As examples, our results contain generating functions of the Al-Salam-Carlitz (I and II), little q-Laguerre, q-Laguerre, and discrete q-Hermite (I and II) polynomials. Our results are more complete and general than those of Refs. 20 and 21 by Chihara. The following are special cases of our results in each class. Here {α_n, ω_n} are the Jacobi-Szegö parameters.

We will prove that some weighted graphs on the distance-k graph of hypercubes approximate the q-Hermite polynomial of a q-gaussian variable by providing an appropriate matrix model.

The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands.

Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property.

The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like.

In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.

In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term |u_t|^q–2u_t and a source term of the type|u|^p–2u. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and show that the solution is global for q ≥ p. Secondly, by an appropriate energy inequality, the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q.

We consider spatial population dynamics given by Markov birth-and-death process with constant mortality and birth influenced by establishment or fecundity mechanisms. The independent and density dependent dispersion of spreading are studied. On the base of general methods of Ref. 14, we construct the state evolution of considered microscopic ecological systems. We analyze mesoscopic limit for stochastic dynamics under consideration. The corresponding Vlasov-type non-linear kinetic equations are derived and studied.

The time evolution of one mode interacting field which interacts with a vee-configuration three-level atom has been discussed. Generated nonclassical state has been utilized to study population inversion of atom and the total noise of the system. Also nonclassicality of the system is shown with the help of Mandel's parameter.

We propose a definition of cycle representation for Quantum Markov Semigroups (QMS) and of Quantum Entropy Production Rate (QEPR) in terms of the ρ-adjoint. We introduce the class of circulant QMS, which admit non-equilibrium steady states but exhibit symmetries that allow us to compute explicitly the QEPR, gain a deeper insight into the notion of cycle decomposition and prove that quantum detailed balance holds if and only if the QEPR equals zero.

In this paper, we use the distribution W(x; q) and the Lebesgue's dominated convergence theorem to give a probabilistic derivation of a transformation formula for the Al-Salam-Carlitz polynomials. Some applications of the transformation are also given, which include an extension of the Heine's ₂ϕ₁ transformation formulas and an extension of the Bailey-Daum summation formula.

An algebraic structure for one-dimensional quantum walks is introduced. This structure characterizes, in some sense, one-dimensional quantum walks. A natural computation using this algebraic structure leads us to obtain an effective formula for the characteristic function of the transition probability. Then, the weak limit theorem for the transition probability of quantum walks is deduced by using simple properties of the Chebyshev polynomials.

We prove a stochastic Gronwall lemma of the following type: if Z is an adapted non-negative continuous process which satisfies a linear integral inequality with an added continuous local martingale M and a process H on the right-hand side, then for any p ∈ (0, 1) the pth moment of the supremum of Z is bounded by a constant κ_p (which does not depend on M) times the pth moment of the supremum of H. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant c_p appearing in the inequality which is at most four times as large as the optimal constant.