Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes

Iteration of randomly chosen quadratic maps defines a Markov process: X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with values in the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Harris recurrence of X_n.Comment: Published at http://dx.doi.org/10.1214/105051604000000918 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Large sample theory of intrinsic and extrinsic sample means on manifolds--II

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere S^d (directional spaces), real projective space RP^{N-1} (axial spaces), complex projective space CP^{k-2} (planar shape spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space \Sigma_3^4.Comment: Published at http://dx.doi.org/10.1214/009053605000000093 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ground State of Quantum Jahn-Teller Model: Selftrapping vs Correlated Phonon-assisted Tunneling

Ground state of the quantum Jahn-Teller model with broken rotational symmetry was investigated by the variational approach in two cases: a lattice and a local ones. Both cases differ by the way of accounting for the nonlinearity hidden in the reflection-symmetric Hamiltonian. In spite of that the ground state energy in both cases shows the same features: there appear two regions of model parameters governing the ground state: the region of dominating selftrapping modified by the quantum effects and the region of dominating phonon-assisted tunneling (antiselftrapping). In the local case (i) the effect of quantum fluctuations and anharmonicity due to the two-mode correlations is up to two orders larger than contributions due to the reflection effects of two-center wave function; (ii) the variational results for the ground state energy were compared with exact numerical results. The coincidence is the better the more far away from the transition region at the E$\otimes$e symmetry where the variational approach fails.Comment: 18 pages, 7 figures, published in Adv.in Quantum Chemistr

Darboux transformation for two-level systems

We develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two-level system. Thus, we find three classes of new solutions for the two-level system and the corresponding new potentials that allow such solutions.Comment: 10 page

Random Iterates of Monotone Maps

In this paper we prove the existence, uniqueness and stability of the invariant distribution of a random dynamical system in which the admissible family of laws of motion consists of monotone maps from a closed subset of a finite dimensional Euclidean space into itself.

Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes

Iteration of randomly chosen quadtratic maps defines a Markov process: X[subscript n + 1] = epsilon[subscript n + 1] X[subscript n](1 - X[subscript n]), where epsilon[subscript n] are i.i.d. with values in the parameter space [0, 4] of quadratic maps F[subscript theta](x) = theta*x(1 - x). Its study is of significance not only as an important Markov model, but also for dynamical systems defined by the individual quadratic maps themselves. In this article a broad criterion is established for positive Harris recurrence of X[subscript n], whose invariant probability may be viewed as an approximation to the so-called Kolmogorov measure of a dynamical system.