Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.     

Multidimensional random subshifts of finite type and their large deviations

Summary I introduce random multidimensional subshifts of finite type which generalize models of spin-glasses and establish the “almost sure” large deviations bounds for Gibbs measures there. The paper is sequel to [EKW] where the corresponding results were obtained for deterministic multidimensional subshifts of finite type. Partially supported by US-Israel BSF     

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities.     

Dimension of hyperbolic measures of random diffeomorphisms

We consider dynamics of compositions of stationary random diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.

     

On phase transitions for subshifts of finite type

It has recently been shown that a strongly irreducible subshift of finite type in two or more dimensions may have more than one measure of maximal entropy. In this paper we obtain some results on when (i.e. for what kinds of subshifts of finite type) this happens, and when it does not. In particular, we show that the parameter of a certain subshift of finite type introduced by Burton and Steif has a critical value, below which we have a unique measure of maximal entropy, and above which we have non-uniqueness.     

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

Tomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

Hitting and returning to rare events for all alpha-mixing processes

We prove that for any α-mixing stationary process the hitting time of any n-string A_n converges, when suitably normalized, to an exponential law. We identify the normalization constant λ(A_n). A similar statement holds also for the return time.To establish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem of Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any n-string in n consecutive observations goes to zero as n goes to infinity.     

The number of possibilities for random dating

Let G be a regular graph and H a subgraph on the same vertex set. We give surprisingly compact formulas for the number of copies of H one expects to find in a random subgraph of G.     

Distributional chaos in random dynamical systems

In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.     

Bowen-Franks groups of reducible bimodal subshifts of finite type

We study the Bowen-Franks groups of subshifts of finite type associated with reducible bimodal periodic kneading sequences pairs.     

Exponential quasi-dissipativity and boundedness property for switched nonlinear systems

This paper is concerned with exponential quasi-dissipativity for switched nonlinear systems using multiple storage functions and multiple supply rates. First, a new concept of exponential quasi-dissipativity for a switched nonlinear system is proposed. In contrast with dissipative system which does not produce energy in some abstract sense, the supply rate of quasi-dissipativity is the sum of the conventional dissipativity supply rate and the constant supply rate. This means quasi-dissipative system can produce energy by itself. Each active subsystem is only required to be exponentially quasi-dissipative, while the energy changing of each inactive subsystem characterized by cross-supply rates is considered as the transformation of “energy” from the active subsystem. By weakening the conventional dissipation inequality, a broader class of open-loop systems and controllers are admissible, leading to broader application. Second, the ultimate boundedness property is obtained under some constraints on the energy changing of inactive subsystems. Third, the exponential quasi-dissipativity conditions for switched nonlinear systems are obtained by the design of a state-dependent switching law. In particular, the sufficient conditions of exponential quasi-passivity and exponential finite power gain for switched nonlinear systems are given thereby providing generalizations of KYP conditions and HJI inequality. Finally, the effectiveness of the obtained results is verified by two examples.     

Approximation for the Normal Inverse Gaussian Process Using Random Sums

Abstract

We approximate the normal inverse Gaussian (NIG) process with random summations. The random sum we introduce is a random walk subordinated to the first passage time of another independent random walk; the model is interpreted as an internal mechanism at small scale that generates the NIG process. The main result is a functional limit theorem of weak convergence in the Skorohod topology.     

Moments of escape times of random walk

Extending an idea of Spitzer [2], a way to compute the moments of the time of escape from (−N,L) by a symmetric simple random walk is exhibited. It is shown that all these moments depend polynomially onL andN. The research of this author was supported by the National Board of Higher Mathematics, Bombay, India     

Large deviations estimates for self-intersection local times for simple random walk in {\mathbb{Z}}^3

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.      

Limit laws in the generalized random graphs with random vertex weights

In this note we investigate the number of edges and the vertex degree in the generalized random graphs with vertex weights, which are independent and identically distributed random variables.     

Some remarks on the strong law of large numbers for Banach-space valued weakly integrable random variables

The purpose of this paper is to show the equivalence of almost sure convergence of S_n/n, n ≥ 1 and lim sup_n→∞S_n/n < ∞ a.e., where S_n = X₁ + X₂ + … + X_n and X₁, X₂,… are independent identically distributed random elements in a separable Banach space with EX₁ < ∞. This result disproves a result of Pop-Stojanovic [8].     

Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems

The problem of estimating the time-dependent statistical characteristics of a random dynamical system is studied under two different settings. In the first, the system dynamics is governed by a differential equation parameterized by a random parameter, while in the second, this is governed by a differential equation with an underlying parameter sequence characterized by a continuous time Markov chain. We propose, for the first time in the literature, stochastic approximation algorithms for estimating various time-dependent process characteristics of the system. In particular, we provide efficient estimators for quantities such as the mean, variance and distribution of the process at any given time as well as the joint distribution and the autocorrelation coefficient at different times.     

Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications

In this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sine-Gordon equation.