Volume 18, Issue 1

Motivated by the Gaussian channel models (see, for instance [Guo, Shamai and Verdú, IEEE Transactions on Information Theory 51: 1261–1282, 2005]), we calculate the mutual information for processes described by multidimensional stochastic differential equations driven by a multidimensional fractional Brownian motion.

In the present paper, we extend the notion of quantum time shift, and the related results obtained in [Accardi, Barhoumi and Ouerdiane, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9: 215–248, 2006], from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular, this class of processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally, we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way comes somehow as a surprise.

We study existence, uniqueness, semi-group property and a-priori estimates for solutions of backward parabolic Itô equations in domains with boundary. We study also duality between forward and backward equations. The semi-group for backward equations is established in the form of some anti-causality. The novelty is that the semi-group property involves the diffusion term that is a part of the solution.

We consider a model for the distribution of a long homopolymer with an attractive zero-range potential at the origin in (polymer pinning and de-pinning). The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length T of the polymer and a parameter γ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point γ cr = 0 the transition occurs from the globular phase (positive recurrent behavior of the polymer, γ > 0) to the extended phase (Brownian type behavior, γ < 0). The main result of the paper is a detailed analysis of the behavior of the polymer when γ is near γ cr . Our approach is based on analyzing the semigroups generated by the self-adjoint extensions ℒ γ of the Laplacian on parametrized by γ , which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator ℒ γ .