Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper we give the solution of Bessis–Moussa–Villani (BMV) conjecture for the generalized Gaussian random variables

The main examples of generalized Gaussian random variables are q-Gaussian random variables, (-1 ≤ q ≤ 1), related to q-CCR relation and other commutation relations. We will prove that BMV conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function

By using an appropriate space of distributions, , we derive the chaos decomposition property of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure Λ_NB. The constructed decomposition is used to define a nuclear triple of test and generalized functions, where θ is a Young function satisfying some suitable conditions. A general characterization theorems are proven for the Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. By using appropriate renormalization procedure, we obtain the representation of the square of white noise obtained by Accardi–Franz–Skeide in Ref. 5. Finally, we investigate the main aim of this paper which is to give unitary equivalent representations of the Witt algebra in the basis of Pascal white noise theory.

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = p^e(P)q^e'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds:

Let H be a separable Hilbert space, and let ϕ and θ be two strongly commuting CP₀-semigroups on B(H). In a previous paper we constructed a Hilbert space K ⊇ H and two (strongly) commuting E₀-semigroups α and β such that

In this note we prove that if ϕ is not an automorphism semigroup, then the semigroup α (given by the above mentioned construction) is cocycle conjugate to the minimal *-endomorphic dilation of ϕ, and that if ϕ is an automorphism semigroup, then α is also an automorphism semigroup. In particular, we conclude that if ϕ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional), then α is a type I E₀-semigroup.

Let be the generator of a positive (i.e. leaving invariant ) contraction semigroup on , the space of self-adjoint trace class operators on a Hilbert space H, endowed with the trace norm, and let be a positive linear operator such that , . We show that there exists a minimal positive contraction semigroup generated by some and provide a systematic study of the total mass carried by individual trajectories with non-negative initial data ρ. The analysis relies on mathematical properties of two (a priori) different extensions of the functional to .

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall mainly be concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup P_t; further, we identify its infinitesimal generator in the space L² (H; ν).

The aim of this work is to define the infinite tensor product of a countable family {E_i}_{i ∈ 𝕀} of spatial product systems with respect to a family of normalized units, and to analyze the main properties of this construction. Among other things, we show that is an amenable product system, respectively a product system of type I, provided that every product system E_i is amenable, respectively of type I.

Let be a uniformly continuous quantum Markov semigroup on with generator represented in a standard GKSL form and a faithful normal invariant state ρ. In this note we give new algebraic conditions for proving that converges towards a steady state, possibly different from ρ. Indeed, we show that this happens whenever the commutator of (i.e. its fixed point algebra) coincides with the commutator of (where δ_H(X) = [H, X]) for some n ≥ 1. As an application we discuss the convergence to the unique invariant state of a spin chain model.