Infinite Dimensional Analysis, Quantum Probability and Related Topics

There is a famous formula called Lévy's stochastic infinitesimal equation for a stochastic process X(t) expressed in the form

Based on a model for general beam splittings we search for states of boson systems which are invariant under the combination of the evolution given by the splitting procedure and some inherent evolution. It turns out that for finite systems only trivial invariant normal states may appear. However, for locally normal states on a related quasilocal algebra representing states of infinite boson systems, one can find examples of nontrivial invariant states. We consider as example a beam splitting combined with a contraction compensating the loss of intensity caused by the splitting process. In general, we observe interesting connections between the splitting procedure and certain thinning operations in classical probability theory. Several applications to physics seem to be natural since these beam splitting models are used to describe measuring procedures on electromagentic fields.

A rigorous mathematical formulation of higher powers of quantum white noises is given on the basis of the most recent theory of white noise distributions due to Cochran, Kuo and Sengupta. The renormalized quantum Itô formula due to Accardi, Lu and Volovich is derived from the renormalized product formula based on integral kernel operators on white noise functions. During the discussion, the analytic characterization of operator symbols and the expansion theorem for a white noise operator in terms of integral kernel operators are established.

This paper deals with the asymptotic behavior of a quantum dynamical semigroup acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w^*-convergence of to , where is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .

Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SU_q(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Lévy–Khintchine formula.

Let be a unital C^*-subalgebra of the C^*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let denote the convex set of all unital, linear, completely positive and normal maps of into itself. Using Stinespring's theorem, we present a criterion for an element to be extremal. When , this criterion leads to an explicit description of the set of all extreme points of . We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem² that every bistochastic matrix can be expressed as a convex combination of permutation matrices.

We study infinite dimensional stochastic differential equations taking values in a white noise space. We show that under certain assumptions the distribution laws of the solution of such an equation induce generalized functions. The white noise integral equation satisfied by these generalized functions is derived. We apply the results to study the stochastic fluctuation of a two-dimensional neuron.

A large deviation theorem is obtained for a certain sequence of random measures which includes the empirical eigenvalue distribution of Wishart matrices, as the matrix size tends to infinity. The rate function is convex and one of its ingredients is the logarithmic energy. In the case of the singular Wishart matrix, the limit distribution has an atom.

We give a characterization of pure finitely correlated states (quantum Markov states) as zero energy states of UHF algebras.

We prove that any probability measure on ℝ, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the nonsymmetric case) a function of the number operator. This follows from a canonical isomorphism between the L²-space of the measure and the interacting Fock space in which the number vectors go into the orthogonal polynomials of the measure and the modified field operator into the multiplication operator by the x-coordinate. A corollary of this is that all the momenta of such a measure are expressible in terms of the Szëgo–Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only noncrossing pair partitions (and singletons, in the nonsymmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call Gaussianization. Finally we define, in terms of the Szëgo–Jacobi parameters, a new convolution among probability measures which we call universal because any probability measure (with all moments) is infinitely divisible with respect to this convolution. All these results will be extended to the case of many (in fact infinitely many) variables in a future paper.