Existence Of Invariant Measure Research Articles

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

A family of robust chaotic S-unimodal maps based on the Gaussian function

This research paper introduces a family of one-dimensional S-unimodal maps based on the Gaussian function, designed to exhibit robust chaos across a wide range of parameters. These maps are developed to display robust chaos by avoiding multiple fixed points that are primarily responsible for the coexisting attractors in 1D maps. The parameter space analysis reveals that chaotic behaviour is sustained across the entire parameter space, except for a very narrow region. The study employs a comprehensive computational approach, including quantitative measures such as sample entropy, Lyapunov exponent, and invariant measures. The uniformly higher values of sample entropy, uniform positive values of the Lyapunov exponent, and the existence of invariant measures in a region of parameter space confirm the presence of robust chaos in these maps. Such a promising class of robust chaotic maps may be potentially used in diverse fields such as chaos-based cryptography, pseudo-random number generation, communication systems, and more.

Persistence in Perturbed Contact Models in Continuum

Can a local disaster lead to extinction? We answer this question in this work. In the paper [19] we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated suf- ficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in [19] was the critical regime condition, which meant the existence of a balance between birth and death rates in average. In the present work, we reject the criticality condition and suppose that the bal- ance condition is violated. This implies that the evolution of the correlation functions of the contact model under consideration is determined by a nonlocal convolution type operator perturbed by a (negative) potential. We show that local peaks in mortality do not typically lead to extinction. We prove that a family of invariant measures exists even without the criticality condition and these measures can be described using the Feynman-Kac formula.

Dynamics of locally monotone stochastic evolution equations

This paper is concerned with the dynamics of the locally monotone stochastic evolution equations in Bochner spaces. We first prove the well-posedness of the abstract stochastic evolution equations with locally Lipschitz nonlinear diffusion terms. Then we establish a mean random dynamical system and prove the existence and uniqueness of weak pullback mean random attractors generated by the mean random dynamical system. Furthermore, we prove the existence of invariant measures for the locally monotone model. As applications, the related dynamics for several stochastic models such as stochastic semilinear equations, stochastic Burgers equation and stochastic 2D Navier-Stokes equations are established.

Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on Rn.

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

Weak random periodic solutions of random dynamical systems

AbstractWe first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their relationship with weak random periodic solutions. In particular, we establish the existence of invariant measures of random dynamical systems by virtue of their weak random periodic solutions. We use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.

Stochastic Doubly Nonlinear PDE: Large Deviation Principles and Existence of Invariant Measure

Stochastic Doubly Nonlinear PDE: Large Deviation Principles and Existence of Invariant Measure

Ergodic properties of a semilinear partial differential equation

Semiflows on some spaces of continuous functions generated by a semilinear partial differential equation are studied. Sufficient conditions for the existence of invariant measures with strong ergodic properties are given. These properties imply chaotic behaviour of the semiflows, that is, the existence of dense trajectories and that each trajectory is unstable. In the proofs we use the Lévy d-parameter Brownian motion and we show that these semiflows are isomorphic with translation along the radii.

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

AbstractThis paper is mainly concerned with the long‐term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville‐type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.

Invariant measure for 2D stochastic Cahn–Hilliard–Navier–Stokes equations

In this paper, we investigate the stochastic Cahn–Hilliard–Navier–Stokes equations in two-dimensional spaces. Applying the Maslowski–Seidler method, we establish the existence of invariant measure in state space [Formula: see text] with the weak topology. We also prove the existence of global pathwise solutions using the stochastic compactness argument.

Asymptotic behavior of stochastic functional differential evolution equation

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.

Entropies for factor maps of amenable group actions

In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles (Zhu 2021 Nonlinearity 34 5163–85, theorems 2.12 and 3.9) proved by Zhu under additional assumptions. Then along the line of Misiurewicz (1976 Studia Math. 55 175–200), we introduce a new invariant called relative topological tail entropy and prove a Ledrappier’s type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

We consider a 2D stochastic modified Swift–Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space for each . Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on . At last, we improve the regularity of the obtained invariant measures to . With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz et al in their situation (2014 J. Math. Phys. 55 277–304).

Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation.

Existence of Invariant Measure and Stationary Equilibrium in a Continuous-Time One-Asset Aiyagari Model: A Case of Regular Controls Under Markov Chain Uncertainty

Existence of Invariant Measure and Stationary Equilibrium in a Continuous-Time One-Asset Aiyagari Model: A Case of Regular Controls Under Markov Chain Uncertainty

Random attractors and invariant measures for 3D stochastic globally modified Navier-Stokes equations with time-dependent delay and coefficient

In this article, we investigate the dynamics of three-dimensional, non-autonomous, and globally modified Navier-Stokes equations with time-dependent delay and additive noise. Based on the well-posedness of solutions, we demonstrate that the solution operators of such equations generate a non-autonomous random dynamical system. We further prove the existence and uniqueness of random attractors as well as the periodicity and the upper semicontinuity for this random dynamical system. We also prove the existence of invariant measures supported on the random attractors. Moreover, we rigorously find that every limit of a sequence of invariant measures must be an invariant measure of the corresponding limiting equations as the noise intensity $ \varepsilon\rightarrow0 $ or the modification parameter $ N\rightarrow N_0\in (0, +\infty) $.

Relative topological conditional entropy and a Ledrappier's type variational principle for it

In this paper we provide a sufficient condition for the existence of invariant measures with maximal relative measure-theoretic entropy, by introducing a new invariant for any factor map between topological dynamical systems, the concept of relative topological conditional entropy. It is proved a Ledrappier's type variational principle concerning relative topological conditional entropy. Consequently, if the factor map has zero relative topological conditional entropy (such a factor map is called asymptotically $ h $-expansive), then there exist invariant measures with maximal relative measure-theoretic entropy (i.e., whose relative measure-theoretic entropy equals exactly relative topological conditional entropy of the factor map). We explore further properties of relative topological conditional entropy, for example, the addition formula for a product factor map, the estimation of it for composition of two factor maps, and so on. We also interpret relative topological conditional entropy, respectively, using relative topological entropy of subsets and Bowen's dimensional entropy.

A stochastic delayed SIS epidemic model with Holling type II incidence rate

In this article, a stochastic SIS epidemic model with constant time delay and Holling type II incidence rate is investigated. We firstly show the existence, uniqueness, and moment boundedness of the global positive solution. Then we extend the initial value space to a complete nonnegative continuous function space and obtain the existence of invariant measures for this system. Furthermore, the analysis of the asymptotic behavior around the disease-free equilibrium is given. To demonstrate, some numerical examples are provided to illustrate our results.

Ergodicity and approximations of invariant measures for stochastic lattice systems with Markovian switching

This paper is concerned with the dynamics of stochastic lattice systems with Markovian switching. Based on the well-posedness of solutions, we first prove the ergodicity of invariant measures and show that the Markov chain facilitates the existence of invariant measures. In order to investigate numerical invariant measures, the convergence of invariant measures is considered between the original systems and their finite-dimensional truncated systems. Due to this, we can use the backward Euler-Maruyama method to approximate invariant measures for such infinite-dimensional systems. This work provides a feasible path for the convergence of the finite-dimensional numerical invariant measures to the analytical invariant measure.

Self-Consistent Transfer Operators: Invariant Measures, Convergence to Equilibrium, Linear Response and Control of the Statistical Properties

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented in the paper are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We apply the general statements to examples of different nature: coupled continuous maps, coupled expanding maps, coupled systems with additive noise, systems made of different maps coupled by a mean field interaction and other examples of self consistent transfer operators not coming from coupled maps. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.