41 research outputs found
Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution
In non relativistic quantum mechanics time enters as a parameter in the
Schroedinger equation. However, there are various situations where the need
arises to view time as a dynamical variable. In this paper we consider the
dynamical role of time through the construction of a Lyapunov variable - i.e.,
a self-adjoint quantum observable whose expectation value varies monotonically
as time increases. It is shown, in a constructive way, that a certain class of
models admit a Lyapunov variable and that the existence of a Lyapunov variable
implies the existence of a transformation mapping the original quantum
mechanical problem to an equivalent irreversible representation. In addition,
it is proved that in the irreversible representation there exists a natural
time ordering observable splitting the Hilbert space at each t>0 into past and
future subspaces.Comment: Accepted for publication in JMP. Supercedes arXiv:0710.3604.
Discussion expanded to include the case of Hamiltonians with an infinitely
degenerate spectru
Nonparametric estimation of the stationary density and the transition density of a Markov chain
In this paper, we study first the problem of nonparametric estimation of the
stationary density of a discrete-time Markov chain . We consider a
collection of projection estimators on finite dimensional linear spaces. We
select an estimator among the collection by minimizing a penalized contrast.
The same technique enables to estimate the density of and
so to provide an adaptive estimator of the transition density . We
give bounds in norm for these estimators and we show that they are
adaptive in the minimax sense over a large class of Besov spaces. Some examples
and simulations are also provided