Ergodic Theory Research Articles

Arcsine Laws of Light.

We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy's arcsine laws-a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).PPRHAI2643-1564]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.

Convergence of Langevin-simulated annealing algorithms with multiplicative noise

We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : R d → R V : \mathbb {R}^d \to \mathbb {R} a potential function to minimize, we consider the stochastic differential equation d Y t = − σ σ ⊤ ∇ V ( Y t ) dY_t = - \sigma \sigma ^\top \nabla V(Y_t) d t + a ( t ) σ ( Y t ) d W t + a ( t ) 2 Υ ( Y t ) d t dt + a(t)\sigma (Y_t)dW_t + a(t)^2\Upsilon (Y_t)dt , where ( W t ) (W_t) is a Brownian motion, where σ : R d → M d ( R ) \sigma : \mathbb {R}^d \to \mathcal {M}_d(\mathbb {R}) is an adaptive (multiplicative) noise, where a : R + → R + a : \mathbb {R}^+ \to \mathbb {R}^+ is a function decreasing to 0 0 and where Υ \Upsilon is a correction term. This setting can be applied to optimization problems arising in Machine Learning; allowing σ \sigma to depend on the position brings faster convergence in comparison with the classical Langevin equation d Y t = − ∇ V ( Y t ) d t + σ d W t dY_t = -\nabla V(Y_t)dt + \sigma dW_t . The case where σ \sigma is a constant matrix has been extensively studied; however little attention has been paid to the general case. We prove the convergence for the L 1 L^1 -Wasserstein distance of Y t Y_t and of the associated Euler scheme Y ¯ t \bar {Y}_t to some measure ν ⋆ \nu ^\star which is supported by argmin ⁡ ( V ) \operatorname {argmin}(V) and give rates of convergence to the instantaneous Gibbs measure ν a ( t ) \nu _{a(t)} of density ∝ exp ⁡ ( − 2 V ( x ) / a ( t ) 2 ) \propto \exp (-2V(x)/a(t)^2) . To do so, we first consider the case where a a is a piecewise constant function. We find again the classical schedule a ( t ) = A log − 1 / 2 ⁡ ( t ) a(t) = A\log ^{-1/2}(t) . We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.

Decision Support System Driven by Thermo-Complexity: Scenario Analysis and Data Visualization

The present modelling aims to construct a computational information representation system useful for decision support system (DSS) solutions in the realization of intelligent systems or complex systems analysis solutions. Starting from an n-dimensional space (with n ≥ 7) represented by problem variables (referred to as CSF—Critical Success Factors), a dimensional embedding procedure is used to transition to a two-dimensional space. In the two-dimensional space, thanks to new lattice motion algorithms, the decision support system can determine the optimal solution with a lower computational cost based on the decision-maker’s preferences. Finally, thanks to an algorithm that takes into account the hierarchical order of importance of the seven CSFs as per the expert’s liking or according to his optimization logics, a return is made to the n-dimensional space and the final solution in the original space. As we will see, the starting and ending states in the n-dimensional space (referred to as micro-states) when projected into the two-dimensional space generate states (referred to as macro-states) which are degenerate. In other words, the correspondence between micro-states and macro-states is not one-to-one, as multiple micro-states correspond to one macro-state. Therefore, in relation to the decision-maker’s preferences, it will be the responsibility of the decision support system to provide the decision-maker with the micro-state of interest in the n-dimensional space (dimensional emergence procedure), starting from the obtained optimal macro-state. This result can be achieved starting from a flat chain of sensors capable of measuring/emulating certain specific parameters of interest. As we will see, it emerges that by considering random–exhaustive rolling value paths in order to track and potentially intervene to rebalance a dynamic system representing the state of stress/sensing of a system of interest, we are using the most general and, therefore, complex hypotheses of ergodic theory. In this work, we will focus on the representation of information in n-dimensional and two-dimensional spaces, as well as construct evaluation scenarios. We will also show the results of the decision support system in some cases of specific interest, thanks to a specific lattice motion algorithm of the realized decision-making environment.

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.

Diffusion of active Brownian particles under quenched disorder.

The motion of a single active particle in one dimension with quenched disorder under the external force is investigated. Within the tailored parameter range, anomalous diffusion that displays weak ergodicity breaking is observed, i.e., non-ergodic subdiffusion and non-ergodic superdiffusion. This non-ergodic anomalous diffusion is analyzed through the time-dependent probability distributions of the particle's velocities and positions. Its origin is attributed to the relative weights of the locked state (predominant in the subdiffusion state) and running state (predominant in the superdiffusion state). These results may contribute to understanding the dynamical behavior of self-propelled particles in nature and the extraordinary response of nonlinear dynamics to the externally biased force.

Random invariant densities for markov operator cocycles and random mean ergodic theorem

In this paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.

On stationary random processes with fuzzy states

In this paper, continuous random processes with fuzzy states are studied. The properties of their numerical characteristics – fuzzy expectations, expected values and covariance functions – are established. The main attention is paid to the class of stationary fuzzy-random processes. For them, the ergodicity property and the spectral representation of covariance function (generalized Wiener–Khinchin theorem) are substantiated. The results obtained are based on the properties of fuzzy-random variables and numerical random processes. Triangular fuzzy-random processes are considered as examples.

Linear progress in fibres

A fibred hyperbolic 3-manifold induces a map from the hyperbolic plane to hyperbolic 3-space, the respective universal covers of the fibre, and the manifold. The induced map is an embedding that is exponentially distorted in terms of the individual metrics. In this article, we begin a study of the distortion along typical rays in the fibre. We verify that a typical ray in the hyperbolic plane makes linear progress in the ambient metric in hyperbolic 3-space. We formulate the proof in terms of some soft aspects of the geometry and basic ergodic theory. This enables us to extend the result to analogous contexts that correspond to certain extensions of closed surface groups. These include surface group extensions that are Gromov hyperbolic, the universal curve over a Teichmüller disc, and the extension induced by the Birman exact sequence.

Strong Ergodicity in Nonhomogeneous Markov Systems with Chronological Order

In the present, we study the problem of strong ergodicity in nonhomogeneous Markov systems. In the first basic theorem, we relax the fundamental assumption present in all studies of asymptotic behavior. That is, the assumption that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix. In addition, we study the practically important problem of the rate of convergence to strong ergodicity for a nonhomogeneous Markov system (NHMS). In a second basic theorem, we provide conditions under which the rate of convergence to strong ergodicity is geometric. With these conditions, we in fact relax the basic assumption present in all previous studies, that is, that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix geometrically fast. Finally, we provide an illustrative application from the area of manpower planning.

An undecidable statement regarding zero-sum games

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist “paradoxical” two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory.

The evolving butterfly: Statistics in a changing attractor

The Earth system is often modeled as a dynamical system in what has come to be known as Earth System Models. When used to study anthropogenically forced climate change, these models are forced in such a way that they are not in a statistically stationary state. Yet, statistical statements are still made about the Earth climate system using only a single trajectory by taking temporal averages. At each moment in time, one draws a sample from a different distribution, raising questions about the utility of temporal averages, in stark contrast to the utility of temporal averages in ergodic systems. This work follows in the tradition of using a toy model to examine properties present in the Earth climate system. We aim to examine how we can make meaningful statistical statements in non-stationary systems when only dealing with a single trajectory. We use the Lorenz equations with a time-varying parameter as a starting point for comparing ensemble averages to temporal averages. We find that, in so far as the control parameter induces a slow and smooth change in the dynamics, the resulting statistics of ensemble averages compare well to those of temporal averages.

Some Generalized Entropy Ergodic Theorems for Nonhomogeneous Hidden Markov Models

Entropy measures the randomness or uncertainty of a stochastic process, and the entropy rate refers to the limit of the time average of entropy. The generalized entropy rate in the form of delayed averages can overcome the redundancy of initial information while ensuring stationarity. Therefore, it has better practical value. A Hidden Markov Model (HMM) contains two stochastic processes, a stochastic process in which all states can be observed and a Markov chain in which all states cannot be observed. The entropy rate is an important characteristic of HMMs. The transition matrix of a homogeneous HMM is unique, while a Nonhomogeneous Hidden Markov Model (NHMM) requires the transition matrices to be dependent on time variables. From the perspective of model structure, NHMMs are novel extensions of homogeneous HMMs. In this paper, the concepts of the generalized entropy rate and NHMMs are defined and fully explained, a strong limit theorem and limit properties of a norm are presented, and then generalized entropy ergodic theorems with an almost surely convergence for NHMMs are obtained. These results provide concise formulas for the computation and estimation of the generalized entropy rate for NHMMs.

Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets’ multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman’s subadditive ergodic theorem.

Most Psychological Researchers Assume Their Samples Are Ergodic: Evidence From a Year of Articles in Three Major Journals

Conventional statistics methods in most psychological research, such as null-hypothesis significance tests (NHSTs), use aggregated values (i.e., the sample means) of group behaviours to make inferences about individuals. Such inferences are possibly erroneous because groups of humans rarely, if ever, constitute an ergodic system. To assume ergodicity without checking is to commit the ‘ergodic fallacy’. The aim of the current study was to examine the prevalence of this error in contemporary psychological research. We analysed three highly cited ‘Q1’ journals in the fields of clinical, educational and cognitive psychology for statements that indicated this error. As hypothesized, the ergodic fallacy was found in the vast majority of the papers investigated here. We also hypothesised that the prevalence of this error would be highest in cognitive psychology papers because this field typically assesses theoretical claims about universal cognitive mechanisms, whereas clinical and educational psychology are more concerned with empirically supported interventions. This hypothesis was also supported by our results. Nonetheless, the prevalence of the ergodic fallacy was still high in all fields. Implications are discussed with respect to the reporting of research findings and the validity of theories in psychology.

A Metric Fixed Point Theorem and Some of Its Applications

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

Summability and speed of convergence in an ergodic theorem

Given an irrational vector α in Rd, a continuous function f(x) on the torus Td and suitable weights Φ(N,n) such that ∑n=−∞+∞Φ(N,n)=1, we estimate the speed of convergence to the integral ∫Tdf(y)dy of the weighted sum ∑n=−∞+∞Φ(N,n)f(x+nα) as N→+∞. Whereas for the arithmetic means N−1∑n=1Nf(x+nα) the speed of convergence is never faster than cN−1, for other means such speed can be accelerated. We estimate the speed of convergence in two theorems with different flavor. The first result is a metric one, and it provides an estimate of the speed of convergence in terms of the Fourier transform of the weights Φ(N,n) and the smoothness of the function f(x) which holds for almost every α. The second result is a deterministic one, and the speed of convergence is estimated also in terms of the Diophantine properties of the given irrational vector α∈Rd.

Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations

Abstract We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.

On the Performance of Noma Systems with Different User Grouping Strategies

This paper investigates several critical issues on the ergodic sum capacity and system outage performance of downlink non-orthogonal multiple access (NOMA) systems with different numbers of users per cluster sharing the same resource. The outcomes from this work suggest that NOMA systems with more users per cluster perform better under a low channel gain, while those with fewer users per cluster perform better under a high channel gain. It discloses a fact that a traditional NOMA system with a fixed user grouping scheme may not be a good choice to deal with complex channel conditions with diverse applications. To address the issue, this paper introduces a concept of mixed user grouping NOMA, which outperforms OMA and traditional NOMA significantly. The work in this paper lays a foundation for more further investigations on mixed user grouping NOMA systems.

Transition to anomalous dynamics in a simple random map.

The famous doubling map (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one, the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability p, and the contracting one with probability 1-p, gives a prototype of a random dynamical system. Here, we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of p. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability pc, defined by a zero Lyapunov exponent. This anomalous dynamics is characterized by an infinite invariant density, weak ergodicity breaking, and power-law correlation decay.

A Mean Ergodic Theorem in Bicomplex Lebesgue Spaces

The main result of this paper is a mean ergodic theorem, in the Von Neumann sense, for some operator acting on the bicomplex Lebesgue space.