Non-ergodic Case Research Articles

Direct characterization of tissue dynamics with laser speckle contrast imaging.

Laser speckle contrast imaging (LSCI) has gained broad appeal as a technique to monitor tissue dynamics (broadly defined to include blood flow dynamics), in part because of its remarkable simplicity. When laser light is backscattered from a tissue, it produces speckle patterns that vary in time. A measure of the speckle field decorrelation time provides information about the tissue dynamics. In conventional LSCI, this measure requires numerical fitting to a specific theoretical model for the field decorrelation. However, this model may not be known a priori, or it may vary over the image field of view. We describe a method to reconstruct the speckle field decorrelation time that is completely model free, provided that the measured speckle dynamics are ergodic. We also extend our approach to allow for the possibility of non-ergodic measurements caused by the presence of a background static speckle field. In both ergodic and non-ergodic cases, our approach accurately retrieves the correlation time without any recourse to numerical fitting and is largely independent of camera exposure time. We apply our method to tissue phantom and in-vivo mouse brain imaging. Our aim is to facilitate and add robustness to LSCI processing methods for potential clinical or pre-clinical applications.

Stationary distributions and ergodicity of reflection-type Markov chains

We consider a reflection-type Markov chain which is usually used to characterize some process features in stochastic control and operations management. The existence of its stationary distributions is established by proving it to be the Feller chain, and its ergodicity is obtained by constructing the small set. Furthermore, for the non-ergodic case, the system dynamics are completely characterized.

ERRATUM: LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: The Nonergodic Case

ERRATUM: LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: The Nonergodic Case

Inference for fractional Ornstein-Uhlenbeck type processes with periodic mean in the non-ergodic case

In the paper we consider the problem of estimating parameters entering the drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case, when the underlying stochastic integral is of Young type. We consider the sampling scheme that the process is observed continuously on and For known Hurst parameter i.e. the long range dependent case, we construct a least-squares type estimator and establish strong consistency. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic function with a rate depending on H and a non-central Cauchy limit result for the mean reverting parameter with exponential rate. For the special case that the periodicity parameter is the weight of a periodic function, which integrates to zero over the period, we can even improve the rate to

Bowen entropy of sets of generic points for fixed-point free flows

Let (X,ϕ) be a compact metric flow without fixed points. We will be concerned with the entropy of flows which takes into consideration all possible reparametrizations of flows. In this paper, by establishing the Brin–Katok's entropy formula for flows without fixed points in the non-ergodic case, we prove the following result: for an ergodic ϕ-invariant measure μ,htopB(ϕ,Gμ(ϕ))=hμ(ϕ1), where Gμ(ϕ) is the set of generic points for μ and htopB(ϕ,Gμ(ϕ)) is the Bowen entropy on Gμ(ϕ). This extends the classical result of Bowen in 1973 to fixed-point free flows. Moreover, it is shown that the Bowen entropy can be determined via the local entropies of measures.

Efficiency Fluctuations of Stochastic Machines Undergoing a Phase Transition.

We study the efficiency fluctuations of a stochastic heat engine made of N interacting unicyclic machines and undergoing a phase transition in the macroscopic limit. Depending on N and on the observation time, the machine can explore its whole phase space or not. This affects the engine efficiency that either strongly fluctuates on a large interval of equiprobable efficiencies (ergodic case) or fluctuates close to several most likely values (nonergodic case). We also provide a proof that despite the phase transition, the decay rate of the efficiency distribution at the reversible efficiency remains largest one although other efficiencies can now decay equally fast.

Parameter estimation for Gaussian mean-reverting Ornstein–Uhlenbeck processes of the second kind: Non-ergodic case

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind [Formula: see text] defined as [Formula: see text], with unknown parameters [Formula: see text] and [Formula: see text], where [Formula: see text] with [Formula: see text], and [Formula: see text] is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of [Formula: see text] and [Formula: see text] based on the continuous-time observations [Formula: see text] as [Formula: see text], we impose some technical conditions on the process [Formula: see text], which are satisfied, for instance, if [Formula: see text] is a fractional Brownian motion with Hurst parameter [Formula: see text], [Formula: see text] is a subfractional Brownian motion with Hurst parameter [Formula: see text] or [Formula: see text] is a bifractional Brownian motion with Hurst parameters [Formula: see text]. Our method is based on pathwise properties of [Formula: see text] and [Formula: see text] proved in the sequel.

Parameter estimation for certain nonstationary processes driven by α-stable motions

The present article deals with the problem of parametric estimation for certain nonstationary processes driven by α-stable motions. The trajectory fitting method combined with the weighted least squares technique is used to obtain an estimator of the drift parameter. The asymptotic behavior of the weighted trajectory fitting estimator for general weights in non-ergodic case, including consistency and asymptotic distribution.

LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: The NonErgodic Case

We formulate and study the infinite dimensional linear programming (LP) problem associated with the deterministic discrete time long-run average criterion optimal control problem. Along with its du...

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(\alpha -\beta X_t)\,dt+\gamma \,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $\alpha$ and $\beta$ in the non-ergodic case (when $\beta <0$) for arbitrary $x_0\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=\alpha /\beta$, derive their asymptotic distributions and prove their asymptotic independence.

Schwinger–Dyson equations and line integrals

The complex Langevin (CL) method sometimes shows convergence to the wrong limit, even though the Schwinger–Dyson equations (SDE) are fulfilled. We analyze this problem in a more general context for the case of one complex variable. We prove a theorem that shows that under rather general conditions not only the equilibrium measure of CL but any linear functional satisfying the SDE on a space of test functions is given by a linear combination of integrals along paths connecting the zeroes of the underlying measure and noncontractible closed paths. This proves rigorously a conjecture stated long ago by one of us (L. L. S.) and explains a fact observed in nonergodic cases of CL.

On limit theorems for fields of martingale differences

We prove a central limit theorem for stationary multiple (random) fields of martingale differences f∘Ti̲, i̲∈Zd, where Ti̲ is a Zd action. In most cases the multiple (random) fields of martingale differences is given by a completely commuting filtration. A central limit theorem proving convergence to a normal law has been known for Bernoulli random fields and in Volný (2015) this result was extended to random fields where one of generating transformations is ergodic.In the present paper it is proved that a convergence takes place always and the limit law is a mixture of normal laws. If the Zd action is ergodic and d≥2, the limit law need not be normal.For proving the result mentioned above, a generalisation of McLeish’s CLT for arrays (Xn,i) of martingale differences is used. More precisely, sufficient conditions for a CLT are found in the case when the sums ∑iXn,i2 converge only in distribution.The CLT is followed by a weak invariance principle. It is shown that central limit theorems and invariance principles using martingale approximation remain valid in the non-ergodic case.

Least squares estimation for the drift parameters in the sub-fractional Vasicek processes

While the statistical inference of Vasicek processes driven by both Brownian motions and fractional Brownian motions has a long history, the statistical analysis for the Vasicek model driven by other fractional Gaussian processes is obviously more recent. This paper considers the parameter estimation problem for Vasicek processes driven by sub-fractional Brownian motions with the known Hurst parameter greater than one half. Since the maximum likelihood estimators are hardly analyzed because of the stochastic integrals with singular kernels, least squares estimators for drift parameters are provided based on time-continuous observations. The strong consistency results as well as the asymptotic distributions of these estimators are obtained in both the non-ergodic case and the null recurrent case.

Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.

Quantifying nonergodicity in nonautonomous dissipative dynamical systems: An application to climate change.

In nonautonomous dynamical systems, like in climate dynamics, an ensemble of trajectories initiated in the remote past defines a unique probability distribution, the natural measure of a snapshot attractor, for any instant of time, but this distribution typically changes in time. In cases with an aperiodic driving, temporal averages taken along a single trajectory would differ from the corresponding ensemble averages even in the infinite-time limit: ergodicity does not hold. It is worth considering this difference, which we call the nonergodic mismatch, by taking time windows of finite length for temporal averaging. We point out that the probability distribution of the nonergodic mismatch is qualitatively different in ergodic and nonergodic cases: its average is zero and typically nonzero, respectively. A main conclusion is that the difference of the average from zero, which we call the bias, is a useful measure of nonergodicity, for any window length. In contrast, the standard deviation of the nonergodic mismatch, which characterizes the spread between different realizations, exhibits a power-law decrease with increasing window length in both ergodic and nonergodic cases, and this implies that temporal and ensemble averages differ in dynamical systems with finite window lengths. It is the average modulus of the nonergodic mismatch, which we call the ergodicity deficit, that represents the expected deviation from fulfilling the equality of temporal and ensemble averages. As an important finding, we demonstrate that the ergodicity deficit cannot be reduced arbitrarily in nonergodic systems. We illustrate via a conceptual climate model that the nonergodic framework may be useful in Earth system dynamics, within which we propose the measure of nonergodicity, i.e., the bias, as an order-parameter-like quantifier of climate change.

Nonergodic complexity management.

Linear response theory, the backbone of nonequilibrium statistical physics, has recently been extended to explain how and why nonergodic renewal processes are insensitive to simple perturbations, such as in habituation. It was established that a permanent correlation results between an external stimulus and the response of a complex system generating nonergodic renewal processes, when the stimulus is a similar nonergodic process. This is the principle of complexity management, whose proof relies on ensemble distribution functions. Herein we extend the proof to the nonergodic case using time averages and a single time series, hence making it usable in real life situations where ensemble averages cannot be performed because of the very nature of the complex systems being studied.

Maximum likelihood estimation for Wishart processes

In the last decade, there has been a growing interest to use Wishart processes for modeling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli (2014) and are of independent interest.

On full groups of non-ergodic probability-measure-preserving equivalence relations

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

Intrinsically weighted means and non-ergodic marked point processes

Mean marks form a versatile toolbox in the analysis of marked point processes (MPPs). For ergodic processes, their definition is straightforward and practical application is well established. In the stationary non-ergodic case, though, different definitions of mark averages are possible and might be practically relevant. In this paper, the classical definition of mean marks is compared to a set of new characteristics for non-ergodic MPPs, which stand out due to the weighting of ergodicity classes. Another weighting can be introduced on the single-point level via weights given by the marks themselves. These intrinsically given weights and the weighting of ergodicity classes are closely related to each other meaning that for suitable choices of weights, a mean mark characteristic can be expressed in either way. Estimators for the different definitions of mean marks are discussed and their consistency and asymptotic normality are shown under certain conditions.

Second-order continuous-time non-stationary Gaussian autoregression

The objective of the paper is to identify and investigate all possible types of asymptotic behavior for the maximum likelihood estimators of the unknown parameters in the second-order linear stochastic ordinary differential equation driven by Gaussian white noise. The emphasis is on the non-ergodic case, when the roots of the corresponding characteristic equation are not both in the left half-plane.