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The Functional Central Limit Theorem for Ergodic Markov Processes
In this chapter a broadly applicable functional central limit theorem for continuous parameter ergodic Markov processes is developed. This uses the...
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Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes
We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is...
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Stationary Ergodic Markov Processes: SLLN & FCLT
For discrete parameter Markov processes on a general state space, Birkhoff’s ergodic theorem provides a natural approach to the existence of...
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A Zero-One Law for the Rates of Convergence in the Birkhoff Ergodic Theorem with Continuous Time
AbstractWe consider monotone pointwise estimates for the rates of convergence in the Birkhoff ergodic theorem with continuous time. For an ergodic...
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Quasi-Similarity, Entropy and Disjointness of Ergodic Actions
AbstractWe answer a question posed by Vershik regarding connections between quasi-similarity of dynamical systems and Kolmogorov entropy. We prove...
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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,...
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Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators
In this survey, we give an introduction to and proof of the gap labelling theorem for discrete one-dimensional ergodic Schrödinger operators via the...
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Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets
We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin...
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Towards mesoscopic ergodic theory
The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic differential equations (SDEs) with...
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The Central Limit Theorem for Markov Processes that are Exponentially Ergodic in the Bounded-Lipschitz Norm
In this paper, we establish a version of the central limit theorem for Markov–Feller continuous time processes (with a Polish state space) that are...
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Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures
Let (Ω, μ ) be a measurable space with a σ -finite continuous measure, μ (Ω) = ∞. A linear operator T : L 1 (Ω) + L ∞ (Ω) → L 1 (Ω) + L ∞ (Ω) is called a Dunford–...
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On an Ergodic Two-Sided Singular Control Problem
Motivated by applications in natural resource management, risk management, and finance, this paper is focused on an ergodic two-sided singular...
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Spectra of Self-Similar Ergodic Actions
AbstractSelf-similar constructions of transformations preserving a sigma-finite measure are considered and their properties and the spectra of the...
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Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical Theory with Transformative Interdisciplinary Impact
Ergodic theory emerged as a statistical mechanics hypothesis and has quickly reached into a mature and influential mathematical theory. Beginning...
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Polynomial averages and pointwise ergodic theorems on nilpotent groups
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of...
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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 ...