Invariant Measure Research Articles

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Evaluating empathy level amongst the dental students using jefferson scale of physician empathy- health professional students

BackgroundEmpathy is described as one’s ability to perceive and apprehend another person’s feelings, situation, emotions, and problems as their own. Empathetic behavior increases patients’ satisfaction, reduces discomfort, and helps with patient’s satisfaction.ObjectiveTo evaluate the psychometric properties of the Jefferson Empathy Scale and compare the measure of invariance within genders and amongst the public and private sector dental students.MethodThis cross-sectional study utilized JSE-HPS version for research purpose. An exploratory factor analysis was performed to detect underlying factors. Reliability of the study tool was evaluated using Cronbach alpha test. Mann Whitney U test was used to compare the differences in scores between genders and among public and private university students while Student’s t analysis compared the scores according to different domains. The level of significance was ≤ 0.05.ResultsFemales demonstrated higher empathy levels (88.52 ± 14.19) along with private institute students (88.46 ± 13.98). Perspective taking and compassionate care domain was also scored highest by the females (31.73 ± 6.49 & 29.31 ± 6.22) and among second year students (33.30 ± 7.11 & 30.50 ± 7.16). PCA analysis extracted 4 factors namely (i) Health-care-provider’s sense of humor contributed to improved outcome (ii) Health-care provider’s understanding of patients’ feelings and of their families influences treatment outcomes (iii) Understanding body language is as important as verbal communication and (iv) Patients feel better when their feelings are understood, which accounted for the 59.51% of the total variance.ConclusionThe findings revealed that students from private institute and females demonstrated higher empathy score. Moreover, the Jefferson Scale of Empathy (JSE) was found to be a reliable and validated tool for assessment of empathy in our sample population.

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval Tβ(x)=βx (mod 1), x∈[0,1] , β > 0, which are the so-called beta-transformations. For such a random dynamical system, including the case that it is generated by uncountably many maps, we give an explicit formula for the density function of a unique stationary measure under the assumption that the random dynamics is expanding in mean. As an application, in the case that the random dynamics is generated by finitely many maps and the maps are chosen according to a Bernoulli measure, we show that the density function is analytic as a function of parameter in the Bernoulli measure and give its derivative explicitly. Furthermore, for a non-i.i.d. random dynamical system of beta-transformations, we also give an explicit formula for the random densities of a unique absolutely continuous invariant measure under a certain strong expanding condition or under the assumption that the maps randomly chosen are close to the beta-transformation for a non-simple number in the sense of parameter β.

A harmonic property of right invariant priors

AbstractIt is shown that, on any Lie group, the density ratio of the right invariant measure to the left invariant measure is harmonic with respect to the left invariant Riemannian metric. This result is applied to the Bayesian prediction theory on group invariant statistical models. A method of constructing Bayesian prior distributions that asymptotically dominate the right invariant priors is provided.

BECK ANXIETY INVENTORY: MEASUREMENT INVARIANCE, LATENT MEAN COMPARISON, AND RELIABILITY IN ADULTS FROM BUENOS AIRES

This research aimed to provide new psychometric evidence of the Argentine Beck Anxiety Inventory (BAI) version in a sample of adults from Argentina. More specifically, the objectives were: 1) test the factor structure of the BAI; 2) assess gender, education, age, and region invariance; 3) test the latent factor mean difference across gender, education, age, and region; 4) analyze the reliability of the BAI; 5) provide population-based norms. The sample was composed of 1,410 adults ranging between 18 and 65 years (M = 32.41; SD = 10.45). Results showed that the BAI should be interpreted as unidimensional. Full configural, metric, and scalar invariance across gender, education, age, and region were obtained for the unidimensional model. The BAI presented adequate reliability values. Latent mean comparisons showed that men experienced more anxiety than women, that college-educated experienced more anxiety than non-college, that older people experienced more anxiety than younger people, and that people living in Greater Buenos Aires experienced more anxiety than those from Buenos Aires City. The BAI is an invariant measure of anxiety symptoms with good psychometric properties.

Wolbachia invasion dynamics of a random mosquito population model with imperfect maternal transmission and incomplete CI.

In this work, we formulate a random Wolbachia invasion model incorporating the effects of imperfect maternal transmission and incomplete cytoplasmic incompatibility (CI). Under constant environments, we obtain the following results: Firstly, the complete invasion equilibrium of Wolbachia does not exist, and thus the population replacement is not achievable in the case of imperfect maternal transmission; Secondly, imperfect maternal transmission or incomplete CI may obliterate bistability and backward bifurcation, which leads to the failure of Wolbachia invasion, no matter how many infected mosquitoes would be released; Thirdly, the threshold number of the infected mosquitoes to be released would increase with the decrease of the maternal transmission rate or the intensity of CI effect. In random environments, we investigate in detail the Wolbachia invasion dynamics of the random mosquito population model and establish the initial release threshold of infected mosquitoes for successful invasion of Wolbachia into the wild mosquito population. In particular, the existence and stability of invariant probability measures for the establishment and extinction of Wolbachia are determined.

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

We study the zero-noise concentration phenomena of invariant measures for SDE. The models are with locally Lipschitz coefficients and have more than one ergodic state. By the large deviations method and an expression of invariant measures, we estimate the invariant measures in neighborhoods of stable sets, unstable sets and their complements. Our results illustrate that the zero-noise limiting invariant measures will concentrate on the stable sets, where a cost functional W(Ki) is minimized. This implies that the long time behaviors of ODE and SDE must have different judging criteria. Furthermore, we prove the large deviations principle of invariant measures.

Lattice regularization of reduced Kähler-Dirac fermions and connections to chiral fermions

We show how a path integral for reduced Kähler-Dirac fermions suffers from a phase ambiguity associated with the fermion measure that is an analog of the measure problem seen for chiral fermions. However, unlike the case of chiral symmetry, a doubler free lattice action exists which is invariant under the corresponding onsite symmetry. This allows for a clear diagnosis and solution to the problem using mirror fermions resulting in a unique gauge invariant measure. By introducing an appropriate set of Yukawa interactions which are consistent with ’t Hooft anomaly cancellation we conjecture the mirrors can be decoupled from low energy physics. Moreover, the minimal such Kähler-Dirac mirror model yields a light sector which corresponds, in the flat space continuum limit, to the Pati-Salam GUT model.

Similarities and differences between specification and non-uniform specification

Abstract Pavlov [Adv. Math.295 (2016), 250–270; Nonlinearity32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

On the extension of singular linear infinite-dimensional Hamiltonian flows

We study the phenomenon of phase trajectories of a Hamiltonian system going to infinity in a finite time, the phase space of which is a separable Hilbert space. It is shown that if the Hamiltonian is a densely defined quadratic form on the phase space, which is not majorized either from below or from above by the quadratic form of the Hilbert norm, then the phase trajectories allow going to infinity in a finite time. To describe the phase flow of such Hamiltonian systems, an extended phase space is introduced, which is a locally convex space to which the Hamiltonian function, trajectories of the Hamiltonian system, and the symplectic form defined on the original Hilbert space can be extended. Flowinvariant measures on extended space are also studied. The properties of the Koopman unitary representation of the extended phase flow in the Hilbert space of functions that are quadratically integrable with respect to an invariant measure are investigated.

Non-existence of a universal zero-entropy system via generic actions of almost complete growth

Abstract We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

Abstract We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Substreetutions and more on trees

Abstract We define a notion of substitution on colored binary trees that we call substreetution. We show that a point fixed by a substreetution may (or not) be almost periodic, and thus the closure of the orbit under the $\mathbb {F}_{2}^{+}$ -action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measures on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.

An invariant measure of chiral quantum transport

We study the invariant measure of the transport correlator for a chiral Hamiltonian and analyze its properties. The Jacobian of the invariant measure is a function of random phases. Then we distinguish the invariant measure before and after the phase integration. In the former case we found quantum diffusion of fermions and a uniform zero mode that is associated with particle conservation. After the phase integration the transport correlator reveals two types of evolution processes, namely classical diffusion and back-folded random walks. Which one dominates the other depends on the details of the underlying chiral Hamiltonian and may lead either to classical diffusion or to the suppression of diffusion.

Invariant Measures of Stochastic Lattice Plate Equations: Stability, Ergodicity and Mixing

Invariant Measures of Stochastic Lattice Plate Equations: Stability, Ergodicity and Mixing

Invariant measures for random piecewise convex maps

We establish the existence of Lebesgue-equivalent conservative and ergodic σ -finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighbourhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong recurrence to a fixed point.

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

An Approximation to the Invariant Measure of the Limiting Diffusion of G/Ph/n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

In this paper, we develop a stochastic algorithm based on the Euler–Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of [Formula: see text] queues in the Halfin–Whitt regime. Specifically, we prove a nonasymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein’s method for multidimensional diffusions, in which the regularity of Stein’s equation obtained by the partial differential equation (PDE) theory plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of [Formula: see text] queues and its Euler–Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein’s equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role. Funding: X. Jin is supported in part by the Fundamental Research Funds for the Central Universities [Grants JZ2022HGQA0148 and JZ2023HGTA0170]. G. Pang is supported in part by the U.S. National Science Foundation [Grants DMS-1715875 and DMS-2216765]. L. Xu is supported in part by the National Nature Science Foundation of China [Grant 12071499], Macao Special Administrative Region [Grant FDCT 0090/2019/A2], and the University of Macau [Grant MYRG2018-00133-FST]. This work was supported by U.S. National Science Foundation [Grant DMS-2108683].

Dynamical Analysis of Stochastic Predator-prey Model with Scavenger

In this paper, we studied the dynamic properties of predator-prey and scavenger three species system by using ergodic invariant measures. Pengyu Ma. find the five points of dynamical bifurcation of the stochastic model, which happened between extinction and survival of each species. Environmental noise was added and proved by the fact that driving force produced by environmental noise influence the system and it was find that system may extinct or partially extinct. Here, we have analysed the stochastic bifurcation phenomena of the prey-predator with scavenger system from the nature of dynamic bifurcation. The phase plots and time diagram plotted for the different values of parameters. We have verified all the results by numerical simulations.