Infinite Dimensional Analysis, Quantum Probability and Related Topics

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions.

The relationship between the Yang–Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy–Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang–Mills equations and the Lévy–Laplace equation for such Laplacian are not equivalent in contrast to the deterministic case. An equation equivalent to the Yang–Mills equations is obtained. The equation contains the Lévy divergence. It is proved that the Yang–Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of a chiral field. This analogue is also derived using Cèsaro averaging.

We demonstrate the large deviation principle in the small noise limit for the mild solution of semilinear stochastic evolution equations with monotone nonlinearity and multiplicative Poisson noise. A recently developed method in studying the large deviation principle, weak convergent method, is employed. We apply the result obtained by Budhiraja et al.,⁷ that reveals the variational representation of exponential integrals w.r.t. the Poisson random measure. Our framework covers a wide range of parabolic, hyperbolic and functional differential equations. We give some examples to illustrate the applications of our results.

In a fundamental lemma we characterize “generating functions” of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to “unital, associative universal products” on this category, which again define a notion of non-commutative stochastic independence. Using the fundamental lemma, we prove the existence of cumulants and of “cumulant Lie algebras” for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in Muraki’s classification, c-free independence of Bożejko and Speicher, the indented product of Hasebe and the bi-free independence of Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.

We give new and simplified proofs of three basic theorems in the theory of orthogonal polynomials associated to a classical, ${ℝ}^d$-valued random variable $X$ with all moments, namely: (1) The characterization of $X$ in terms of commutators among the creation–annihilation–preservation (CAP) operators in its quantum decomposition. (2) The characterization, in terms of the same objects, of the fact that the distribution of $X$ is a product measure. (3) The equivalence of the symmetry of $X$ with the vanishing of the associated preservation operator.

Our new formulation of these results allows one to obtain a stronger form of the above statements.

In this paper, we study some relevant properties of generic quantum Markov semigroups, in particular related to their asymptotic behavior. We can describe the structure of the set of fixed points and of the invariant states in terms of the Hamiltonian’s spectrum and of the communication classes of the classical Markov process associated with the diagonal part of the semigroup. Moreover we study the decoherence-free algebra and we complete the characterization of environmental decoherence for a generic quantum Markov semigroup.

We prove that all finite joint distributions of creation and annihilation operators in monotone and anti-monotone Fock spaces can be realised as Quantum Central Limit of certain operators in a $C^{*}$-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone $C^{*}$-algebras, the weights being related to the above cited test functions.