Infinite Dimensional Analysis, Quantum Probability and Related Topics

We study ergodicity and decoherence for generic quantum Markov semigroups: in particular, we highlight the strong relationships among these properties, the convergence in law of the classical Markov process associated with the diagonal part of the semigroup, and the structure of its communication classes.

Through the consideration of a homeomorphism between C([0, T] → ℝ^ℤ) and C([0, T] → (S¹)^ℤ), a one-to-one correspondence between stochastic prosesses with values in ℝ^ℤ resp. (S¹)^ℤ is discussed. In particular, uniqueness of solutions of SDE's and contraction properties of the semigroups of the corresponding Markov processes on these spaces are studied.

We consider a sequence of weak Kadison–Schwarz maps τ_n on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τ_n, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (T_n, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ_*.

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under Hörmander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this result, we apply methods of the Malliavin calculus.

We study various ergodic properties of C*-dynamical systems inspired by unique ergodicity. In particular we work in a framework allowing for ergodic properties defined relative to various subspaces, and in terms of weighted means. Our main results are characterizations of such relative ergodic properties, ergodic theorems resulting from these properties, and examples exhibiting these properties.

In this paper, we construct fractional Lévy processes for any parameter H ∈ (0, 1), as the generalization of the fractional Brownian motion. By using Malliavin calculus, we also define the stochastic integral for fractional Lévy processes.

Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20).

In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).

In this work stochastic integration with respect to cylindrical Lévy processes with weak second moments is introduced. It is well known that a deterministic Hilbert–Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Lévy process by random Hilbert–Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic integral of a predictable stochastic process with respect to a cylindrical Lévy process. We finish this work by deriving an Itô isometry and by considering shortly stochastic partial differential equations driven by cylindrical Lévy processes.