Infinite Dimensional Analysis, Quantum Probability and Related Topics

We find the best constant c(m, n, r), such that the inequality:

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

We show that some C*-dynamical systems obtained by free Fock quantization of classical ones, enjoy ergodic properties much stronger than their boson or fermion analogous. Namely, if the classical dynamical system (X, T, μ) is ergodic but not weakly mixing, then the resulting free quantized system (𝔊, α) is uniquely ergodic (w.r.t. the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system (X, T, μ) which is weakly mixing but not mixing. In this case, the free quantized system is uniquely weak mixing but not uniquely mixing. Finally, a free quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing C*-dynamical systems whose Gelfand–Naimark–Segal representation associated to the unique invariant state generates a von Neumann factor of one of the following types: I_∞, II₁, III_λ where λ ∈ (0, 1]. The resulting scenario is then quite different from the classical one. In fact, if a classical system is uniquely mixing, it is conjugate to the trivial one consisting of a singleton. For the sake of completeness, the results listed above are extended to the q-Commutation Relations, provided . The last result has a self-contained meaning as we prove that the involved C*-dynamical systems based on the q-Commutation Relations are conjugate to the corresponding one arising from the free case (i.e. q = 0), at least if .

Let be a quantum Markov semigroup on with a faithful normal invariant state ρ. The decoherence-free subalgebra of is the biggest subalgebra of where the completely positive maps act as homomorphisms. When is the minimal semigroup whose generator is represented in a generalised GKSL form , with possibly unbounded H, L_ℓ, we show that coincides with the generalised commutator of under some natural regularity conditions.

As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and L_ℓ.

We give examples of quantum Markov semigroups , with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

Consider the Lévy–Meixner one-mode interacting Fock space {Γ_LM, 〈 ⋅, ⋅ 〉_LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉_LM, we define scalar products 〈 ⋅, ⋅ 〉_{LM, n} in symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μ_r; r > 0}. The Fourier transform in generalized joint eigenvectors of a family of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘_LM between and the so-called Lévy–Meixner white noise space . We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.

We study the properties of the (noncommutative) bm-independence of algebras, indexed by partially ordered sets. The index sets are given by positive cones, in particular the symmetric cones, which include the positive-definite Hermitian matrices with complex or quaternionic entries. We formulate and prove the general versions of the bm-Central Limit Theorems for bm-independent random variables, indexed by lattices in such positive cones. The limit measures we obtain are symmetric and compactly supported on the real line. Their (even) moment sequences (g_n)_n≥0 satisfy the generalized recurrence for the Catalan numbers: , where the coefficients γ(r) are computed by the Euler's beta-function of the first kind, related to the given positive cone. Example of a nonsymmetric cone, the Vinberg's cone, is also studied in this context.

We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line ℝ is constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson–Watanabe superprocesses, conditioned to have total mass one, are Fleming–Viot superprocesses.

We generalize the notion of subisometric liftings of row contractions for liftings of completely contractive covariant representations of W*-correspondences. A theory of characteristic functions for such liftings of covariant representations is presented.