Rolando Rebolledo

Subharmonic projections for a quantum Markov semigroup

This article introduces a concept of subharmonic projections for a quantum Markov semigroup, in view of characterizing the support projection of a stationary state in terms of the semigroup generator. These results, together with those of our previous article [J. Math. Phys. 42, 1296 (2001)], lead to a method for proving the existence of faithful stationary states. This is often crucial in the analysis of ergodic properties of quantum Markov semigroups. The method is illustrated by applications to physical models.

On the existence of stationary states for quantum dynamical semigroups

We provide two criteria on the existence of stationary states for quantum dynamical semigroups. The first one is based on the semigroup itself, while the second criterion is based on the generator which is in general unbounded and interpreted as a sesquilinear form. These results are illustrated by physical examples drawn from quantum optics.

Quantum flows associated to master equations in quantum optics

A stochastic dilation of master equation for a maser is constructed using the technique of quantum stochastic differential equations with unbounded operator coefficients. The associated quantum Markov flow is characterized and turns out to extend a classical birth and death process. Furthermore, the stationary distribution of the aforementioned process, which is connected to the stationary state of the micromaser, is obtained in a closed form.

THE APPROACH TO EQUILIBRIUM OF A CLASS OF QUANTUM DYNAMICAL SEMIGROUPS

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 .

Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions

The paper is devoted to the study of nonlinear stochastic Schro- dinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born-Markov approximations, this class of stochastic evo- lutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator.

Principles for modelling financial markets

The paper introduces an approach focused towards the modelling of dynamics of financial markets. It is based on the three principles of market clearing, exclusion of instantaneous arbitrage and minimization of increase of arbitrage information. The last principle is equivalent to the minimization of the difference between the risk neutral and the real world probability measures. The application of these principles allows us to identify various market parameters, e.g. the risk-free rate of return. The approach is demonstrated on a simple financial market model, for which the dynamics of a virtual risk-free rate of return can be explicitly computed.

ALGEBRAIC CONDITIONS FOR CONVERGENCE OF A QUANTUM MARKOV SEMIGROUP TO A STEADY STATE

Let

Entropy Production for Quantum Markov Semigroups

{\mathcal T}

THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP WITH UNBOUNDED GENERATOR

be a uniformly continuous quantum Markov semigroup on

REGULARITY OF SOLUTIONS TO LINEAR STOCHASTIC SCHRÖDINGER EQUATIONS

{\mathcal B} ({\mathsf h})