Multiplicative Ergodic Theorem Research Articles

Lyapunov exponents of stochastic differential equations driven by Lévy processes

Lyapunov exponents are studied for stochastic differential equations driven by Lévy processes. The transformation from stochastic differential equations to random integral equations and a multiplicative ergodic theorem are used. Moreover, the result is applied to physical and economic problems.

Conditions for equality between Lyapunov and Morse decompositions

Let$Q\rightarrow X$be a continuous principal bundle whose group$G$is reductive. A flow${\it\phi}$of automorphisms of$Q$endowed with an ergodic probability measure on the compact base space$X$induces two decompositions of the flag bundles associated to$Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

On Lyapunov exponents of difference equations with random delay

The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.

A concise proof of the multiplicative ergodic theorem on Banach spaces

We give a new proof of a multiplicative ergodic theorem for quasi-compactoperators on Banach spaces with a separable dual. Our proof works by constructingthe finite-codimensional `slow' subspaces (those where the growthrate is dominated by some$\lambda_i$), in contrast with earlier infinite-dimensional multiplicative ergodictheorems which work by constructing the finite-dimensional fast subspaces.As an important consequence for applications, we are able to get rid ofthe injectivity requirements that appear in earlier works.

Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.

Reaching Consensus With Increasing Information

This paper focuses on a model of opinion formation over networks with continuously flowing new information and studies the relationship between the network and information structures and agents' ability to reach agreement. At each time period, agents receive private signals in addition to observing the beliefs held by their neighbors in a network. Each agent then updates its belief by aggregating the information available to it in a boundedly rational fashion. Whether this model results in conformity or disparity of beliefs is contingent on connectivity of the network and identifiability of the unknown state. We show that if the network is strongly connected, agents will eventually reach consensus in their beliefs; if in addition the state is globally identifiable, then agents will be able to learn the unknown state. Agents will also reach consensus and learn the state if the network fails to be connected but the state is locally identifiable. In contrast, agents in a non-strongly connected network will almost never reach agreement in their beliefs about a locally unidentifiable state. We also provide a characterization of the rates of convergence in terms of the top Lyapunov exponent of a set of i.i.d. matrices. The proofs use Oseledets' multiplicative ergodic theorem and recent results on stability of Lyapunov regular dynamical systems.

A semi-invertible operator Oseledets theorem

AbstractSemi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.

Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations

In this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in Banach spaces. Such a dichotomy has a tempered bound and describes nonuniform hyperbolicity. We prove the roughness without assuming their invertibility and the integrability condition of the Multiplicative Ergodic Theorem. We give an explicit bound for the linear perturbation such that the dichotomy is persistent. We also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations.

Lyapunov regularity for random dynamical systems

This paper is concerned with the Lyapunov regularity of linear random dynamical systems. Both continuous and discrete cases are treated and the conditions are expressed in terms of the so-called regularity coefficient, which is an essential element of the theory of Lyapunov forward regularity developed by Lyapunov himself. It turns out that, using the celebrated multiplicative ergodic theorem of Oseledets, the Lyapunov forward regularity is “typical” in a measure theoretical sense.

A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc. 19 (1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems 30 (2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.

Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups

Let $Q\rightarrow X$ be a principal bundle having as structuralgroup $G$ areductive Lie group in the Harish-Chandra class that includes the case when $G$ is semi-simple with finite center. A semiflow $\phi _{k}$ ofendomorphisms of $Q$ induces a semiflow $\psi _{k}$ on theassociated bundle $\mathbb{E}=Q\times _{G}\mathbb{F}$, where$\mathbb{F}$ is the maximal flag bundle of $G$. The $A$-componentof the Iwasawa decomposition $G=KAN$ yields an additive vectorvalued cocycle $\mathsf{a}\left( k,\xi \right) $, $\xi\in \mathbb{E}$, over $\psi _{k}$ with values in the Lie algebra $\mathfrak{a}$ of $A$. We prove the Multiplicative Ergodic Theorem ofOseledets for this cocycle: If $\nu $ is a probability measureinvariant by the semiflow on $X$ then the $\mathfrak{a}$-Lyapunovexponent $\lambda \left( \xi \right) =\lim\frac{1}{k}\mathsf{a}\left( k,\xi \right) $ exists for every $\xi$ on the fibers above a set of full $\nu $-measure. The level setsof $\lambda \left( \cdot \right) $ on the fibers are described inalgebraic terms. When $\phi _{k}$ is a flow the description of thelevel sets is sharpened. We relate the cocycle $\mathsf{a}\left(k,\xi \right) $ with the Lyapunov exponents of a linear flow on avector bundle and other growth rates.

Error analysis of a hybrid method for computing Lyapunov exponents

In a previous paper (Beyn and Lust in Numer Math 113:357---375, 2009) we suggested a numerical method for computing all Lyapunov exponents of a dynamical system by spatial integration with respect to an ergodic measure. The method extended an earlier approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223---237, 1999) for the largest Lyapunov exponent by integrating the diagonal entries from the $$QR$$ -decomposition of the Jacobian for an iterated map. In this paper we provide an asymptotic error analysis of the method for the case in which all Lyapunov exponents are simple. We employ Oseledec multiplicative ergodic theorem and impose certain hyperbolicity conditions on the invariant subspaces that belong to neighboring exponents. The resulting error expansion shows that one step of extrapolation is enough to obtain exponential decay of errors.

Almost sure exponential convergence to consensus of random gossip algorithms

SUMMARYIn this paper, we provide a proof of almost sure exponential convergence to consensus for a general class of ergodic edge selection processes. The proof is based on the multiplicative ergodic theorem of Oseledec and also applies to continuous time gossip algorithms. An example of exponential convergence in a non ergodic case is also discussed. Copyright © 2012 John Wiley & Sons, Ltd.

Poisson boundary of GL d (ℚ)

We construct the Poisson boundary for a random walk supported by the general linear group on the rational numbers as the product of flag manifolds over the p-adic fields. For this purpose, we prove a law of large numbers using Oseledets’ multiplicative ergodic theorem. The only assumption we need is some moment condition on the measure governing the jumps of the random walk, but no irreducibility hypothesis is made.

Realization of joint spectral radius via Ergodic theory

Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.

Equidistribution of singular measures on nilmanifolds and skew products

AbstractWe prove that for a minimal rotationTon a two-step nilmanifold and any measureμ, the push-forwardTn⋆μofμunderTntends toward Haar measure if and only ifμprojects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math.91(1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus.Amer. J. Math.83(1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits ofTn⋆μfor some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

BIFURCATIONS IN A STOCHASTIC BUSINESS CYCLE MODEL

We introduce a stochastic business cycle model and study the underlying stochastic Hopf bifurcations with respect to probability densities at different parameter values. Our analysis is based on the calculation of the largest Lyapunov exponent via multiplicative ergodic theorem and the theory of boundary analysis for quasi-nonintegrable Hamiltonian systems. Some numerical simulations of the model are performed.

On index-2 linear implicit difference equations

This paper deals with an index-2 notion for linear implicit difference equations (LIDEs) and with the solvability of initial value problems (IVPs) for index-2 LIDEs. Besides, the cocycle property as well as the multiplicative ergodic theorem of Oseledets type are also proved.

Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space

LYAPUNOV EXPONENTS AND INVARIANT MANIFOLD FOR RANDOM DYNAMICAL SYSTEMS IN A BANACH SPACE Zeng Lian Department of Mathematics Ph.D of Mathematics Abstract We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

The asymptotic shape theorem for generalized first passage percolation

We generalize the asymptotic shape theorem in first passage percolation on $\mathbb{Z}^d$ to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for $\mathbb{Z}^d$-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon--Nikodym property.