Mean Ergodicity Research Articles

On the speed of convergence in the ergodic theorem for shift operators

Abstract Given a probability space $(X,\mu )$ , a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means $N^{-1}\sum _{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$ . We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document}, for t∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1)$$\\end{document}, are investigated when they act on the disc algebra A(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A({\\mathbb {D}})$$\\end{document} and on the Hardy spaces Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document}, for 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}. We study the continuity, compactness, spectrum and point spectrum of Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} as well as their linear dynamics and mean ergodicity on these spaces.

Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces

In this paper, we introduce inductive limits of the Fréchet spaces ℓ(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell (p+)$$\\end{document}, ces(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {ces}(p+)$$\\end{document}, and d(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(p+)$$\\end{document} (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p < \\infty $$\\end{document}) and projective limits of the (LB)-spaces ℓ(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell (p-)$$\\end{document}, ces(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {ces}(p-)$$\\end{document}, and d(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(p-)$$\\end{document} (1<p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < p \\le \\infty $$\\end{document}). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} (0≤t≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le t \\le 1$$\\end{document}) act continuously in these sequence spaces, and we determine the spectra. Finally, we study the ergodic properties, that is, power boundedness, (uniform) mean ergodicity, and supercyclicity, of the operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} acting in the (LF)-spaces and in the (PLB)-spaces mentioned above.

Ergodic mean field games: existence of local minimizers up to the Sobolev critical case

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

Ergodicity for a family of operators

The aim of this paper is to introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of bounded linear operators on a Banach space. The authors present some elementary results in this setting and show that some main results about power bounded, Cesàro bounded, mean ergodic, and the uniform ergodic operator can be extended from the case of a linear bounded operator to the case of a family of bounded linear operators acting on a Banach space. Also, we show that the Yosida theorem can be extended from the case of a bounded linear operator to the case of a family of bounded linear operators acting on a Banach space.

Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

AbstractAn investigation is made of the generalized Cesàro operators $$C_t$$ C t , for $$t\in [0,1]$$ t ∈ [ 0 , 1 ] , when they act on the space $$H({{\mathbb {D}}})$$ H ( D ) of holomorphic functions on the open unit disc $${{\mathbb {D}}}$$ D , on the Banach space $$H^\infty $$ H ∞ of bounded analytic functions and on the weighted Banach spaces $$H_v^\infty $$ H v ∞ and $$H_v^0$$ H v 0 with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of $$C_t$$ C t as well as their linear dynamics and mean ergodicity.

Ergodic properties of multiplication and weighted composition operators on spaces of holomorphic functions

AbstractWe obtain different results about the mean ergodicity of weighted composition operators when acting on the spaces , , and , where is the open unit ball of a Banach space, as well as about the compactness and the mean ergodicity of the multiplication operator. This study relates these properties of the operators with properties of the symbol and the weight defining such operators.

Seemingly unrelated time series model for forecasting the peak and short-term electricity demand: Evidence from the Kalman filtered Monte Carlo method

In this extant paper, a multivariate time series model using the seemingly unrelated times series equation (SUTSE) framework is proposed to forecast the peak and short-term electricity demand using time series data from February 2, 2014, to August 2, 2018. Further the Markov Chain Monte Carlo (MCMC) method, Gibbs Sampler, together with the Kalman Filter were applied to the SUTSE model to simulate the variances to predict the next day's peak and electricity demand. Relying on the study results, the running ergodic mean showed the convergence of the MCMC process. Before forecasting the peak and short-term electricity demand, a week's prediction from the 28th to the 2nd of August of 2018 was analyzed and it found that there is a possible decrease in the daily energy over time. Further, the forecast for the next day (August 3, 2018) was about 2187 MW and 44090 MWh for the peak and electricity demands respectively. Finally, the robustness of the SUTSE model was assessed in comparison to the SUTSE model without MCMC. Evidently, SUTSE with the MCMC method had recorded an accuracy of about 96% and 95.8% for Peak demand and daily energy respectively.

Mean Ergodic Weighted Shifts on Köthe Echelon Spaces

Necessary and sufficient conditions are given for mean ergodicity, power boundedness, and topologizability for weighted backward shift and weighted forward shift operators, respectively, on Köthe echelon spaces in terms of the weight sequence and the Köthe matrix. These conditions are evaluated for the special case of power series spaces which allow for a characterization of said properties in many cases. In order to demonstrate the applicability of our conditions, we study the above properties for several classical operators on certain function spaces.

Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators C_t, for tin [0,1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}, such as ell ^p, c_0, c, bv_0, bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesàro spaces ces(p) and their (isomorphic) dual spaces d_p. In most cases C_t (tnot =1) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of C_t, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}. Besides {{mathbb {C}}}^{{{mathbb {N}}}_0} itself, the Fréchet spaces considered are ell (p+), ces(p+) and d(p+), for 1le p<infty , as well as the (LB)-spaces ell (p-), ces(p-) and d(p-), for 1<ple infty .

Almost sure contraction for diffusions on [formula omitted]. Application to generalized Langevin diffusions

In the case of diffusions on Rd with constant diffusion matrix, without assuming reversibility nor hypoellipticity, we prove that the contractivity of the deterministic drift is equivalent to the constant rate contraction of Wasserstein distances Wp, p∈[1,∞]. It also implies concentration inequalities for ergodic means of the process. Such a contractivity property is then established for some non-equilibrium chains of anharmonic oscillators and for some generalized Langevin diffusions when the potential is convex with bounded Hessian and the friction is sufficiently high. This extends previous known results for the usual (kinetic) Langevin diffusion.

Stationary Discounted and Ergodic Mean Field Games with Singular Controls

We study stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and ergodic performance criteria. This class of games finds natural applications in the context of optimal productivity expansion in dynamic oligopolies. We prove the existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations. Furthermore, we relate the mean field equilibria for the discounted and ergodic games by showing the validity of an Abelian limit. The latter also allows us to approximate Nash equilibria of—so far unexplored—symmetric N-player ergodic singular control games through the mean field equilibrium of the discounted game. Numerical examples finally illustrate in a case study the dependency of the mean field equilibria with respect to the parameters of the games. Funding: The authors acknowledge financial support from the Alan Turing Institute and CMAP, École Polytechnique during this project. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 317210226 – SFB 1283.

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesàro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.

On diagonal operators between the sequence (LF)-spaces l_p({mathcal {V}})

Diagonal (multiplication) operators acting between a particular class of countable inductive spectra of Fréchet sequence spaces, called sequence (LF)-spaces, are investigated. We prove results concerning boundedness, compactness, power boundedness, and mean ergodicity. Furthermore, we determine when a diagonal operator is Montel and reflexive. We analyze the spectra in terms of the system of weights defining the spaces.

Dynamics of composition operators on function spaces defined by local and global properties

In this paper we consider composition operators on locally convex spaces of functions defined on R. We prove results concerning supercyclicity, power boundedness, mean ergodicity and convergence of the iterates in the strong operator topology.

Mean ergodic composition operators on spaces of smooth functions and distributions

We investigate (uniform) mean ergodicity of weighted composition operators on the space of smooth functions and the space of distributions, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic with period 2. Our results are based on a characterization of mean ergodicity in terms of Cesàro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest.

Mean ergodicity of multiplication operators on the Bloch and Besov spaces

Mean ergodicity of multiplication operators on the Bloch and Besov spaces

Mean ergodicity of Toeplitz operators on Hardy spaces

Mean ergodicity of Toeplitz operators on Hardy spaces

Spectrums and Uniform Mean Ergodicity of Weighted Composition Operators on Fock Spaces

For holomorphic pairs of symbols (u, psi ), we study various structures of the weighted composition operator W_{(u,psi )} f= u cdot f(psi ) defined on the Fock spaces mathcal {F}_p. We have identified operators W_{(u,psi )} that have power-bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values |u(0)| and |u(frac{b}{1-a})|, where a and b are coefficients from linear expansion of the symbol psi . The spectrum of the operators is also determined and applied further to prove results about uniform mean ergodicity.

An ergodic correspondence principle, invariant means and applications

A theorem due to Hindman states that if E is a subset of ℕ with d*(E) > 0, where d* denotes the upper Banach density, then for any ε > 0 there exists N ∈ ℕ such that $$d^{\ast}(\cup_{i=1}^{N}(E-i))>1-\varepsilon$$ . Curiously, this result does not hold if one replaces the upper Banach density d* with the upper density $$\bar{d}$$ . Originally proved combinatorially, Hindman’s theorem allows for a quick and easy proof using an ergodic version of Furstenberg’s correspondence principle. In this paper, we establish a variant of the ergodic Furstenberg’s correspondence principle for general amenable (semi)-groups and obtain some new applications, which include a refinement and a generalization of Hindman’s theorem and a characterization of countable amenable minimally almost periodic groups.