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Under suitable hypotheses we establish a quantitative pointwise ergodic theorem which applies to trimmed Birkhoff sums of weakly integrable functions.

Dynamical Systems · Mathematics 2019-02-20 Alan Haynes

The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a…

Probability · Mathematics 2017-04-10 Nicolas Champagnat , Denis Villemonais

We survey some recent developments and give a list of open problems regarding multiple recurrence and convergence phenomena of $\mathbb{Z}^d$ actions in ergodic theory and related applications in combinatorics and number theory.

Dynamical Systems · Mathematics 2016-10-18 Nikos Frantzikinakis

We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling…

Statistical Mechanics · Physics 2015-05-14 Attilio L. Stella , Fulvio Baldovin

Many classical objects of study related to the geometry/topology of smooth Gaussian fields (e.g., the volume, surface area or Euler characteristic of excursion sets) have a `locality' property which is crucial to their analysis. More…

Probability · Mathematics 2026-02-26 Michael McAuley

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. We consider the corresponding eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the ergodic…

Dynamical Systems · Mathematics 2014-07-28 Xavier Bressaud , Alexander I. Bufetov , Pascal Hubert

We introduce and study a dimensional-like characteristic of an uniformly almost periodic function, which we call the Diophantine dimension. By definition, it is the exponent in the asymptotic behavior of the inclusio length. Diophantine…

Dynamical Systems · Mathematics 2017-10-10 Mikhail Anikushin

We obtain limit theorems (Stable Laws and Central Limit Theorems, both Gaussian and non-Gaussian) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The proofs of the limit…

Dynamical Systems · Mathematics 2020-03-24 Henk Bruin , Dalia Terhesiu , Mike Todd

We consider when there is absolute or unconditional convergence of series of various types of stochastic processes. These processes include differences of averages in ergodic theory and harmonic analysis, like the classical Cesaro average…

Dynamical Systems · Mathematics 2025-01-17 Bryan Johnson , Joseph Rosenblatt

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we…

Number Theory · Mathematics 2008-09-11 Roman Holowinsky

We obtain pointwise ergodic theorems with rate under conditions expressed in terms of the convergence of series involving $\|\sum_{k=1} ^nf\circ \theta^k\|_2$, improving previous results. Then, using known results on martingale…

Probability · Mathematics 2009-04-02 Christophe Cuny

The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier--Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a…

Analysis of PDEs · Mathematics 2021-05-19 Francesco Fanelli , Eduard Feireisl , Martina Hofmanová

A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies…

Statistics Theory · Mathematics 2021-11-30 Morgane Austern , Peter Orbanz

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal…

Combinatorics · Mathematics 2024-01-09 Vitaly Bergelson , Rigoberto Zelada

The purpose of this article is to discuss the circle method and its quantitative role in understanding pointwise almost everywhere convergence phenomena for polynomial ergodic averaging operators. Specifically, we will use the circle method…

Dynamical Systems · Mathematics 2026-02-13 Mariusz Mirek

We extend to Lipschitz continuous functionals either of the true paths or of the Euler scheme with decreasing step of a wide class of Brownian ergodic diffusions, the Central Limit Theorems formally established for their marginal empirical…

Probability · Mathematics 2013-04-03 Gilles Pagès , Fabien Panloup

In recent years, the ergodic theory of group actions on homogeneous spaces has played a significant role in the metric theory of Diophantine approximation. We survey some recent developments with special emphasis on Diophantine properties…

Number Theory · Mathematics 2016-06-09 Anish Ghosh

We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…

Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum…

Quantum Physics · Physics 2022-07-29 Ramis Movassagh , Jeffrey Schenker

We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone, and almost…

Mathematical Physics · Physics 2018-09-28 Christoph Schumacher , Fabian Schwarzenberger , Ivan Veselic
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