Narrow Results

We study the invariant measures in the weak disorder limit, for the Anderson model on two coupled chains. These measures live on a three-dimensional projective space, and we use a total set of functions on this space to characterize the measures. We find that at several points of the spectrum, there are anomalies similar to that first found by Kappus and Wegner for the single chain at zero energy.

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied.

In finite-dimensional dissipative dynamical systems, stochastic stability provides the selection of the physically relevant measures. That this might also apply to systems defined by partial differential equations, both dissipative and conservative, is the inspiration for this work. As an example, the 2D Euler equation is studied. Among other results this study suggests that the coherent structures observed in 2D hydrodynamics are associated with configurations that maximize stochastically stable measures uniquely determined by the boundary conditions in dynamical space.

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest nontrivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.

The main purpose of this tutorial is to introduce to a more application-oriented audience a new chaos theory that is applicable to certain systems of differential equations. This new chaos theory, namely the theory of rank one maps, claims a comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain systems of differential equations, the existence of the indicated phenomenon of chaos can be verified through a well-defined computational process. Applications to the well-known Chua's and MLC circuits employing controlled switches are also presented to demonstrate the usefulness of the theory. We try to introduce this new chaos theory by using a balanced combination of examples, numerical simulations and theoretical discussions. We also try to create a standard reference for this theory that will hopefully be accessible to a more application-oriented audience.

It has been suggested that the properties of "integration" and "differentiation" are necessary for the emergence of consciousness. We present a dynamical system model that is based on these two conditions. The collection of neurons are partitioned into clusters on which we define a map that reflects the communication between clusters. Such a map displays the forward and backward circuitry between clusters in a probabilistic manner. The presence of "re-entry" guarantees that the map is sufficiently complex, that is, nonlinear and chaotic, to possess numerous invariant sets of clusters, which are referred to as agglomerations. We suggest that an agglomeration that is mixing characterizes a conscious state. The model establishes a theoretical framework that may structure and encourage experimental work.

Let {τ₁, τ₂,…,τ_K} be a collection of nonsingular maps on [0, 1] into [0, 1] and {p₁, p₂,…,p_K} be a collection of position dependent probabilities on [0, 1]. We consider position dependent random maps T = {τ₁,τ₂,…,τ_K;p₁,p₂,…,p_K} such that T preserves an absolutely continuous invariant measure with density f*. A maximum entropy method for approximating f* is developed. We present a proof of convergence of the maximum entropy method for random maps.

We present a numerical method for the approximation of absolutely continuous invariant measures of one-dimensional random maps, based on the maximum entropy principle and piecewise linear moment functions. Numerical results are also presented to show the convergence of the algorithm.

We analyze the structure of the frequency spaceQ(F) of a nonabelian free group F = F(a₁,…,a_k) consisting of all shift-invariant Borel probability measures on ∂F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, where w is a nontrivial conjugacy class, is the intersection of ℚ and a closed subinterval of ℝ with rational endpoints.

In this paper we study some properties of the polytope of belief functions on a finite referential. These properties can be used in the problem of identification of a belief function from sample data. More concretely, we study the set of isometries, the set of invariant measures and the adjacency structure. From these results, we prove that the polytope of belief functions is not an order polytope if the referential has more than two elements. Similar results are obtained for plausibility functions.

In this paper we study the long-time behaviour a stochastic parabolic equation perturbed through the boundary; the perturbation is represented by a nonhomogeneous Dirichlet boundary condition of white-noise type. The existence of the solution for an equation of this kind was the object of a previous paper of the same authors. The estimates proved therein allow to study the asymptotic properties of the solution; we show that there exists a unique invariant measure that is exponentially mean square stable.

The transition semigroup corresponding to stochastic obstacle problem is irreducible.

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall mainly be concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup P_t; further, we identify its infinitesimal generator in the space L² (H; ν).

We consider the linear stochastic Cauchy problem

In this paper we study a particular class of forward rate problems, related to the Vasicek model, where the driving equation is a linear Gaussian stochastic partial differential equation. We first give an existence and uniqueness results of the related mild solution in infinite dimensional setting, then we study the related Ornstein–Uhlenbeck semigroup with respect to the determination of a unique invariant measure for the associated Heath–Jarrow–Morton–Musiela model.

We prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$th moment of solutions depends on the integrability of the initial datum, in the whole range $p\in ]0,\infty [$. Lipschitz continuity of the solution map in $p$th moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range $p\in [0,\infty [$. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.

We prove that the Gibbs measures $\rho$ for a class of Hamiltonian equations written as

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

Invariant measure-type notions are examined for nonautonomous dynamical systems, namely, for flows whose dynamics is affected by an external parameter. The desired measures are obtained as projections of invariant measures of the shift flow on the lifted space of trajectories. The specific cases of multi-valued dynamics and controlled dynamics are examined in detail and other applications are pointed out.

We study the existence of strong solutions for a class of stochastic differential equations in an infinite dimensional space. Our investigation is specially motivated by the stochastic version of a common model of potential spread in a dendritic tree. We do not assume the noise in the junction points to be Markovian. In fact, we allow for long-range dependence in time of the stochastic perturbation. This leads to an abstract formulation in terms of a stochastic diffusion with dynamic boundary conditions, featuring fractional Brownian motion. We prove results on existence, uniqueness and asymptotics of weak and strong solutions to such a stochastic differential equation.