Exponential Mixing Property Research Articles

Dynamics of Ostrowski skew-product: 1. Limit laws and Hausdorff dimensions

We present a dynamical study of Ostrowski’s map based on the use of transfer operators. The Ostrowski dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and intervals as fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operators, we show that the absolutely continuous invariant measure of the Ostrowski dynamical system has exponential mixing properties. We deduce a central limit theorem for random variables of an arithmetic nature, and motivated by applications in inhomogeneous Diophantine approximation, we also get Bowen–Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a bounded digit set.

Existence, exponential mixing and convergence of periodic measures of fractional stochastic delay reaction-diffusion equations on [formula omitted

This paper is concerned with periodic measures of fractional stochastic reaction-diffusion equations with variable time delay defined on unbounded domains. We first prove the existence of periodic measures of the Markov process associated with the equation by showing the weak compactness of distribution laws of a family of solutions based on the uniform estimates and the equicontinuity of solutions in probability. We then establish the regularity of periodic measures and prove the tightness of the set of all periodic measures of the equation for small noise intensity. As a result, any sequence of periodic measures possesses at least one limit point, and we further prove every such limit must be a periodic measure of the corresponding limiting system. Finally, under further assumptions on the nonlinear terms, we establish the uniqueness and the exponential mixing property of periodic measures in the sense of Wasserstein metric. All the results of the paper are also valid for invariant measures of the corresponding autonomous version of the stochastic equation with constant time delay. The idea of uniform tail estimates of solutions in probability is employed to overcome the difficulty introduced by the non-compactness of Sobolev embeddings on unbounded domains.

On the continuity of topological entropy of certain partially hyperbolic diffeomorphisms

In this paper, we consider certain partially hyperbolic diffeomorphisms (PHDs) with centre of arbitrary dimension and obtain continuity properties of the topological entropy under C 1 perturbations. The systems considered have subexponential growth in the centre direction and uniform exponential growth along the unstable foliation. Our result applies to PHDs which are Lyapunov stable in the centre direction. It applies to another important class of systems which do have subexponential growth in the centre direction, for which we develop a technique to use exponential mixing property of the systems to get uniform distribution of unstable manifolds. A primary example are the translations on homogeneous spaces which may have centre of arbitrary dimension and of polynomial orbit growth.

A mechanical true random number generator

Random number generation has become an indispensable part of information processing: it is essential for many numerical algorithms, security applications, and in securing fairness in everyday life. Random number generators (RNGs) find application in many devices, ranging from dice and roulette wheels, via computer algorithms, lasers to quantum systems, which inevitably capitalize on their physical dynamics at respective spatio-temporal scales. Herein, to the best of our knowledge, we propose the first mathematically proven true RNG (TRNG) based on a mechanical system, particularly the triple linkage of Thurston and Weeks. By using certain parameters, its free motion has been proven to be an Anosov flow, from which we can show that it has an exponential mixing property and structural stability. We contend that this mechanical Anosov flow can be used as a TRNG, which requires that the random number should be unpredictable, irreproducible, robust against the inevitable noise seen in physical implementations, and the resulting distribution’s controllability (an important consideration in practice). We investigate the proposed system’s properties both theoretically and numerically based on the above four perspectives. Further, we confirm that the random bits numerically generated pass the standard statistical tests for random bits.

Exponential mixing property for Hénon–Sibony maps of

AbstractLet f be a Hénon–Sibony map, also known as a regular polynomial automorphism of $\mathbb {C}^k$ , and let $\mu $ be the equilibrium measure of f. In this paper we prove that $\mu $ is exponentially mixing for plurisubharmonic observables.

Exponential mixing property for absorbing Markov processes

In this paper, we study the speed of mixing of absorbing Markov processes with state space E, that is, the rate of the convergence in the following limits: limt→∞Ex[f(Xpt)g(Xt)|T>t]=∫Ef(y)β(dy)∫Eg(y)α(dy),limt→∞Ex[f(Xpt)g(Xqt)|T>t]=∫Ef(y)β(dy)∫Eg(y)β(dy),where f,g are bounded and measurable functions defined on E, x∈E, 0<p<q<1, T is the absorption time of the process, α is a quasi-stationary distribution and β is a quasi-ergodic distribution. Under general conditions, we show that the convergence rates of the limits are exponential. As an application, we apply our result to the logistic Feller diffusion process and the birth–death process with catastrophes.

Well-posedness and exponential mixing for stochastic magneto-hydrodynamic equations with fractional dissipations

Consider d-dimensional magneto-hydrodynamic (MHD) equations with fractional dissipations driven by multiplicative noise. First, we prove the existence of martingale solutions for stochastic fractional MHD equations in the case of d = 2, 3 and α ⋀ β > 0, where α, β are the parameters of the fractional dissipations in the equation. Second, for d = 2, 3 and $$\alpha \wedge \beta \geqslant {1 \over 2} + {d \over 4}$$ , we show the pathwise uniqueness of solutions and then obtain the existence and uniqueness of strong solutions using the Yamada-Watanabe theorem. Furthermore, we establish the exponential mixing property for stochastic MHD equations with degenerate multiplicative noise when d = 2, 3 and $$\alpha \wedge \beta \geqslant {1 \over 2} + {d \over 4}$$ .

Exponential mixing property for automorphisms of compact Kähler manifolds

Exponential mixing property for automorphisms of compact Kähler manifolds

Exponential mixing properties of the stochastic tamed 3D Navier–Stokes equation with degenerate noise

The current paper is devoted to the stochastic tamed 3D Naiver-Stokes equation with degenerate noise with degenerate random forcing. We prove that the associated Markov semigroup possesses an exponentially attracting invariant measure by showing that the corresponding dynamical system possesses a strong type of Lyapunov structure along with a special type of gradient inequality and is of a relatively weak form of irreducibility.

Exponential mixing for stochastic 3D fractional Leray-[formula omitted] model with degenerate multiplicative noise

In this work we establish the exponential mixing property for stochastic 3D fractional Leray-α model with degenerate noise. The main result is also applicable to stochastic 3D hyperviscous Navier–Stokes equations and stochastic 3D critical Leray-α model.

Long-time behavior of 3D stochastic planetary geostrophic viscous model

In this paper, we consider the 3D planetary geostrophic viscous model driven by an additive random noise. It is proved that this model has exponential mixing property and global random attractor at the same time. Further, we deduce that the support of the integration of the invariant measure for the dynamic generated by this model is exactly a minimal global random attractor. Moreover, we show that the global random attractor has finite Hausdorff dimension.

Effective mixing and counting in Bruhat–Tits trees

Let ${\mathcal{T}}$ be a locally finite tree, $\unicode[STIX]{x1D6E4}$ be a discrete subgroup of $\,\operatorname{Aut}\,({\mathcal{T}})$ and $\widetilde{F}$ be a $\unicode[STIX]{x1D6E4}$-invariant potential. Suppose that the length spectrum of $\unicode[STIX]{x1D6E4}$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map $\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}\rightarrow \unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}$ with respect to the measure $m_{\unicode[STIX]{x1D6E4},F}^{\unicode[STIX]{x1D708}^{-},\unicode[STIX]{x1D708}^{+}}$ under the assumption that $\unicode[STIX]{x1D6E4}$ is full and $(\unicode[STIX]{x1D6E4},\widetilde{F})$ has a weighted spectral gap. We also obtain the effective formula for the number of $\unicode[STIX]{x1D6E4}$-orbits with weights in a Bruhat–Tits tree ${\mathcal{T}}$ of an algebraic group.

Exponential mixing properties for time inhomogeneous diffusion processes with killing

We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^d$ with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion semigroup; when a particle is absorbed, another selected particle splits into two offsprings. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.

Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$

In this note, we consider the orbits $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ in $\Gamma\backslash\text{PSL}(2,\mathbb R)$, where $\Gamma$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent group in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point $p$, we can prove that $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ is equidistributed in $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, which generalizes a result of Venkatesh (Ann.of Math. 2010). We will compute Hausdorff dimensions of subsets of non-Diophantine points in Appendix A, using results of lattice counting problem. In Appendix B we will use a technique of Venkatesh (Ann.of Math. 2010) and an exponential mixing property to prove a weak version of a result of Str\"ombergsson (J Mod Dynam, 2013), which is about the effective equidistribution of horospherical orbits.

Exponential mixing for stochastic model of two-dimensional second grade fluids

In this paper, we establish the exponential mixing property of stochastic model for the incompressible second grade fluids. The general criterion established by Odasso (2008) plays an important role.

Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral points on sections of quadrics. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.

Quasi-likelihood analysis for the stochastic differential equation with jumps

In this paper, we consider a multidimensional diffusion process X with jumps whose jump term is driven by a compound Poisson process, and discuss its parametric estimation. We present asymptotic normality and convergence of moments of any order for a quasi-maximum likelihood estimator and a Bayes type estimator by assuming an exponential mixing property of X. To show these properties, we use the polynomial type large deviation theory.

Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions

We study an infinite white $\alpha$-stable systems with unbounded interactions, proving the existence by Galerkin approximation and exponential mixing property by an $\alpha$-stable version of gradient bounds.

Continuity properties of transport coefficients in simple maps

We consider families of dynamics that can be described in terms of Perron–Frobenius operators with exponential mixing properties. For piecewise C2 expanding interval maps we rigorously prove continuity properties of the drift J(λ) and of the diffusion coefficient D(λ) under parameter variation. Our main result is that D(λ) has a modulus of continuity of order , i.e. D(λ) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are quantified numerically for the latter class of maps by using exact series expansions for the transport coefficients that can be evaluated numerically. We numerically observe strong local variations of all continuity properties.

Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model

With the help of a general methodology of asymptotic expansions for mixing processes, we obtain the Edgeworth expansion for log-returns of a stock price process in Barndorff-Nielsen and Shephard's stochastic volatility model, in which the latent volatility process is described by a stationary non-Gaussian Ornstein–Uhlenbeck process (OU process) with invariant selfdecomposable distribution on R + . The present result enables us to simultaneously explain non-Gaussianity for short time-lags as well as approximate Gaussianity for long time-lags. The Malliavin calculus formulated by Bichteler, Gravereaux and Jacod for processes with jumps and the exponential mixing property of the OU process play substantial roles in order to ensure a conditional type Cramér condition under a certain truncation. Owing to several inherent properties of OU processes, the regularity conditions for the expansions can be verified without any difficulty, and the coefficients of the expansions up to any order can be explicitly computed.