Some Ergodic Theorems for Random Rotations on Wiener Space

A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multip…

Algebraic structure of the $L_2$ analytic Fourier-Feynman transform associated with Gaussian processes on Wiener space

In this paper we study algebraic structures of the classes of the $L_2$ analytic Fourier-Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian processes. We then proceed to analyze the $L_2$ analytic Fourier-Feynman transforms associated with Gaussian processes. Our results show that these $L_2$ analytic Fourier--Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener…

吉本斯-兰道方程在二维环面上与复值时空白噪声的遍历性

Ergodicity for Ginzburg-Landau equation with complex-valued space-time white noise on two-dimensional torus

研究复吉本斯-兰道方程在二维环面上的全局可解性及遍历性，采用固定点定理和时间衰减估计证明全局可解性，利用克里洛-博戈留波夫定理和渐进行耦合论证证明遍历性，使用复多重维纳-伊托积分理论估计随机分布。

Ergodicity in Riesz spaces

The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Rie…

对数遍历过程的若干应用：遍历交易模型与利用无理旋转的看涨期权定价

Some Applications of Log-Ergodic Processes: Ergodic Trading Model and Call Option Pricing Using the Irrational Rotation

针对期货风险管理，本文构建股票离场时机模型，并利用遍历定理、单位圆上的随机无理旋转及对数遍历过程的时间平均，研究欧式看涨期权定价。

Diffuse Behaviour of Ergodic Sums Over Rotations

For a rotation by an irrational $α$ on the circle and a BV function $φ$, we study the variance of the ergodic sums $S_L φ(x) := \sum_{j=0}^{L -1} \, φ(x + jα)$. When $α$ is not of constant type, we construct sequences $(L_N)$ such that, at some scale, the ergodic sums $S_{L_N} φ$ satisfy an ASIP. Explicit non-degenerate examples are given, with an application to the rectangular periodic billiard in the plane.

On random almost periodic trigonometric polynomials and applications to ergodic theory

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(λ_k^{(1)}t_1+...+λ_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{λ_n^{(i)}:1\leq i\leq s\}$ are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered $\{X_n\}$ or bounded $\{X_n\}$ satisfying some mixing conditions. These results generalize recent results of Weber [Math. Inequal. Appl. 3 (2000) 443--457] and Fan and Schneider [Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 193--216] in several directions. As applications we derive c…

On a nonsymmetric Ornstein-Uhlenbeck semigroup and its generator

If we add a simple rotation term to both the Ornstein-Uhlenbeck semigroup and the definition of the H-derivative, then analogue to the classical Malliavin calculus on the real Wiener space [I. Shigekawa, Stochastic analysis, 2004], we get a normal but nonsymmetric Ornstein-Uhlenbeck operator $L$ on the complex Wiener space. The eigenfunctions of the operator $L$ are given. In addition, the hypercontractivity for the nonsymmetric Ornstein-Uhlenbeck semigroup is shown.

Multi-valued, singular stochastic evolution inclusions

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-* mean ergodic invariant measure.

关于局部单调随机演化方程的遍历性与混合性

Ergodicity and mixing for locally monotone stochastic evolution equations

本文研究了局部单调驱动和退化加性Wiener噪声的随机演化方程，建立了存在唯一不变概率测度的定量条件，并提供了解的矩估计和半群的e-性质，以及2-Wassersteinε-混合时间的上界。

Rational ergodicity of Step function Skew Products

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with Z_q-invariant Bernoulli-increments which has as local state space the cyclic group Z_q. We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S^1. The update mechanism we construct is exponentially well localized on the lattice.

Equidistribution of random walks on compact groups II. The Wasserstein metric

We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with Hölder continuous test functions $f$, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided the distribution of $S_k$ converges to the Haar measure in the $p$-Wasserstein metric fast enough. As an example we construct discrete random walks on an irrational lattice on the torus $\mathbb{R}^d/\mathbb{Z}^d$, and find their precise rate o…

强Feller马尔可夫过程在子集内经验测度的大偏差与拟遍历分布

Large deviations of the empirical measures of a strong-Feller Markov process inside a subset and quasi-ergodic distribution

本文针对强Feller过程建立了子区域内不退出条件下占用测度的大偏差原理，确定速率函数仅在拟遍历分布处消失且可逆时为狄利克雷形式，并将结果应用于椭圆随机微分方程解等多个随机过程。

大旋转极限下随机原始方程的平均化原理

Averaging principle for the stochastic primitive equations in the large rotation limit

本文证明带加性白噪声的粘性原始方程在大旋转极限下，强解的分布被吸引到随机极限共振系统的唯一不变测度，从而表现出近乎唯一的遍历性。

Ergodic behaviour of nonconventional ergodic averages for commuting transformations

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to the space average, of diagonal measures is obtained and we give different result according to the classification of transformations. Additionally, on the torus with special rotation. we can not only get the convergence in T.Tao's paper for every point in Td, but also get a beautiful result for ergodic behaviour.

Ergodicity for stochastic equation of Navier--Stokes type

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier--Stokes equations system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant probability measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite-dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the …

A Sampling Theorem for Rotation Numbers of Linear Processes in ${\R}^{2}$

We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^{1}$. In particular, the concept of rotation number of a matrix $g\in Gl^{+}(2,{\R})$ can be generalized to a product of a sequence of stationary random matrices in $Gl^{+}(2,{\R})$. In this particular case this result provides a counter-part of the Osseledec's multiplicative ergodic theorem which guarantees the existence of Lyapunov exponents. A random sampling theorem is then proved to show that the concept we propose is consistent by discretization in time with the …

Ergodicity of stochastic shell models driven by pure jump noise

In the present paper we study a stochastic evolution equation for shell (SABRA \& GOY) models with pure jump \levy noise $L=\sum_{k=1}^\infty l_k(t)e_k$ on a Hilbert space $\h$. Here $\{l_k, k\in \mathbb{N}\}$ is a family of independent and identically distributed (i.i.d.) real-valued pure jump \levy processes and $\{e_k, k\in \mathbb{N}\}$ is an orthonormal basis of $\h$. We mainly prove that the stochastic system has a unique invariant measure. For this aim we show that if the \levy measure of each component $l_k(t)$ of $L$ satisfies a certain order and a non-{degeneracy} condition and is ab…

遍历泛函中心极限定理的离散化

Discretization of the Ergodic Functional Central Limit Theorem

本文研究平稳遍历马尔可夫过程泛函中心极限定理的离散化，提出 q 阶时间积分离散方法，证明 CLT 仍成立但速率变为 n^{q/(2q+1)}，并应用于 Euler 格式和 Talay 弱二阶格式逼近布朗扩散平稳态。

Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues

We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…

On the CLT for rotations and BV functions

Let $x \mapsto x+ α$ be a rotation on the circle and let $φ$ be a step function. We denote by $φ\_n (x)$ the corresponding ergodic sums $\sum\_{j=0}^{n-1} φ(x+j α)$. Under an assumption on $α$, for example when $α$ has bounded partial quotients, and a Diophantine condition on the discontinuity points of $φ$, we show that $φ\_n/\|φ\_n\|\_2$ is asymptotically Gaussian for $n$ in a set of density 1. The method is based on decorrelation inequalities for the ergodic sums taken at times $q\_k$, where the $q\_k$'s are the denominators of $α$.

A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hormander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M_t$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $Π$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M_t$ with small eigenvalues have only a very sma…

高温下无限体积 $Φ^4_3$ 的遍历性

Ergodicity of infinite volume $Φ^4_3$ at high temperature

本文研究了无限体积 $Φ^4_3$ 动力学在加权 Besov 空间中的全局适定性，证明高温下解之差指数收敛，从而将无限体积测度表征为唯一不变测度并验证其满足 Osterwalder--Schrader 公理。