Tao Wang
We obtain the lower bounds for ergodic convergence rates, including spectral gaps and convergence rates in strong ergodicity for time-changed symmetric Lévy processes by using harmonic function and reversible measure. As direct applications, explicit sufficient conditions for exponential and strong ergodicity are given. Some examples are also presented.
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In this paper we study the ergodicity and the related semigroup property for a class of symmetric Markov jump processes associated with time changed symmetric $α$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric $α$-stable processes, yield sharp criteria for their various ergodic properties and corre…
We obtain explicit criteria for both exponential ergodicity and strong ergodicity for one-dimensional time-changed symmetric stable processes with $α\in(1,2)$. Explicit lower bounds for ergodic convergence rates are given.
Under proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the M_1-topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed symmetric α-stable Lévy process. The time change is given by the inverse β-stable subordinator.
Quasi-stationary distributions for time-changed symmetric $α$-stable processes killed upon hitting zero
该研究针对从无穷远进入、击中零点即被吸收的时间变换对称$\alpha$-稳定过程，证明了其拟平稳分布的存在唯一性，并在转移密度相关条件下证明任意初始分布指数收敛到该分布。
We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.
We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients $a^ω$. The diffusion is formally associated with $L^ωu = \nabla\cdot(a^ω\nabla u)$, and we make sense of it through Dirichlet forms theory. We prove for $X$ a quenched invariance principle, under some moment conditions on the environment; the key tool is the sublinearity of the corrector obtained by Moser's iteration scheme.
We study a symmetric diffusion process on $\mathbb{R}^d$, $d\geq 2$, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green's function is proven.
We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.
This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite $d\times d$-matrices. In particular, for conservative and subcritical affine processes on this cone we show that a finite $\log$-moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: firstly, in a specific metric induced by the Laplace transform and secondly, in the Wasserstein distance under a first moment assu…
We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.
Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling--Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the Lévy system) for the time-changed process. W…
This paper studies homogenization of symmetric non-local Dirichlet forms with $α$-stable-like jumping kernels in one-parameter stationary ergodic environment. Under suitable conditions, we establish homogenization results and identify the limiting effective Dirichlet forms explicitly. The coefficients of the jumping kernels of Dirichlet forms and symmetrizing measures are allowed to be degenerate and unbounded; and the coefficients in the effective Dirichlet forms can be degenerate.
This is a note on a local ergodic theorem for a symmetric exclusion process defined on an infinite tower of coverings, which is associated with a finitely generated residually finite amenable group.
We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed including diffusion processes on Riemannian manifolds and Ornstein-Uhlenbeck processes driven by symmetric $α$-stable processes. For SDE driven by $α$-stable process ($α\in (0,2]$) with polynomial drift, the strong ergodicity or not is independent on $α$.
We prove the existence of a successful coupling for $n$ particles in the symmetric inclusion process. As a consequence we characterize the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in the course of time to an invariant product measure.
Let $D$ be an unbounded domain in $\RR^d$ with $d\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on $\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\overline D$. We further show that if there is some $r>0$ so that $D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and…
We establish a connection between the structure of a stationary symmetric alpha-stable random field (0 < alpha < 2) and ergodic theory of non-singular group actions, elaborating on a previous work by Rosinski (2000). With the help of this connection, we study the extreme values of the field over increasing boxes. Depending on the ergodic theoretical and group theoretical structures of the underlying action, we observe different kinds of asymptotic behavior of this sequence of extreme values.
Given the increments of a simple symmetric random walk $(X_n)_{n\ge0}$, we characterize all possible ways of recycling these increments into a simple symmetric random walk $(Y_n)_{n\ge0}$ adapted to the filtration of $(X_n)_{n\ge0}$. We study the long term behavior of a suitably normalized two-dimensional process $((X_n,Y_n))_{n\ge0}$. In particular, we provide necessary and sufficient conditions for the process to converge to a two-dimensional Brownian motion (possibly degenerate). We also discuss cases in which the limit is not Gaussian. Finally, we provide a simple necessary and sufficient …
$L^2$-exponential ergodicity of stochastic Hamiltonian systems with $α$-stable Lévy noises
本文针对带α-稳定Lévy噪声的随机哈密顿系统，通过推广DMS亚强制性框架，在分数阶动力学Fokker-Planck背景下建立不要求算子对称性的对偶版本，从而证明了系统的L²-指数遍历性。
Quasi-ergodic theorems for Feynman-Kac semigroups and large deviation for additive functionals
本文研究了Feynman-Kac半群的准遍历定理和加性函数的大偏差原理，探讨了对称马尔可夫过程的长期行为及相关条件估计。
Generally the convergence rate in exponential ergodicity $λ$ is an upper bound for the convergence rate $κ$ in uniform ergodicity for a Markov process, that is $λ\geqslantκ$. In this paper, we prove that $κ\geqslant \inf \{lambda,1/M_H\}$, where $M_H$ is a uniform bound on the moment of the hitting time to a "compact" set $H$. In the case where $M_H$ can be made arbitrarily small for $H$ large enough, we obtain that $λ=κ$. The general results are applied to Markov chains, diffusion processes and solutions to SDEs driven by symmetric stable processes.
Let $F$ be a non-discrete non-Archimedean locally compact field such that the characteristic $\mathrm{ch}(F)\ne 2$ and let $\mathcal{O}_F$ be the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Skew}(\mathbb{N}, F)$ of infinite skew-symmetric matrices with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.4, that gives an unexpected natural correspondence between the set of $\mathrm{GL}(\infty,\mathcal{O}_F)$-invariant Borel probability measures on $\mathrm{Sym}…
Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset {\mathbb R}^d$ with $N$ number of Liouville branches. We consider a diffusion $X$ on $\overline D$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space ${\rm BL}…
Sampling with time-changed Markov processes
本文提出通过时间变更马尔可夫过程加速MCMC算法收敛，通过状态依赖函数调整基础过程时间速度，证明了其理论性质并在多峰分布和稀有事件模拟中实现性能提升。
This paper elucidates the connection between stationary symmetric alpha-stable processes with 0<alpha<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into those corresponding to positive flows and those corresponding to null flows. We show that a necessary and sufficient for a stationary stable process to be ergodic is that its positive component vanishes.
We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric $α$-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance operator. In particular, in the equation with diagonal pure $α$-stable cylindrical noise introd…
Unique ergodicity of simple symmetric random walks on the circle
本文针对圆周上由无理旋转驱动的简单对称随机游走，给出任意无理角度下唯一平稳分布的一个新证明，简化并推广了 Sinai、Conze 与 Guivarc'h 的经典结果。
This paper is concerned with the approximation of the effective conductivity $σ(A,μ)$ associated to an elliptic operator $\nabla_x A(x,η) \nabla_x$ where for $x\in \R^d$, $d\geq 1$, $A(x,η)$ is a bounded elliptic random symmetric $d\times d$ matrix and $η$ takes value in an ergodic probability space $(X,μ)$. Writing $A^N(x,η)$ the periodization of $A(x,η)$ on the torus $T^d_N$ of dimension $d$ and side $N$ we prove that for $μ$-almost all $η$ $$ \lim_{N\to +\infty}σ(A^N,η)=σ(A,μ) $$ We extend this result to non-symmetric operators $\nabla_x (a+E(x,η)) \nabla_x$ corresponding to diffusions in e…
We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include: a proof of an original averaging principle for stochastic flows of kernels; the definition and study of a convergence o…
Sampling inverse subordinators and subdiffusions
本文提出精确抽样逆从属过程轨迹的方法，适用于一般严格递增从属过程，并研究亚扩散过程泛函的蒙特卡罗近似，证明中心极限定理与强收敛性，应用于弱遍历性破缺。
We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uni…
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…
We consider a driven tagged particle in a symmetric exclusion process on Z with a removal rule. In this process, untagged particles are removed once they jump to the left of the tagged particle. We investigate the behavior of the displacement of the tagged particle and prove limit theorems of it: an (annealed) law of large numbers, a central limit theorem, and a large deviation principle. We also characterize a class of ergodic measures for the environment process. Our approach is based on analyzing two auxiliary processes with associated martingales and a regenerative structure.
We consider symmetric trap models in the d-dimensional hypercube whose ordered mean waiting times, seen as weights of a measure in the natural numbers, converge to a finite measure as d diverges, and show that the models suitably represented converge to a K process as d diverges. We then apply this result to get K processes as the scaling limits of the REM-like trap model and the Random Hopping Times dynamics for the Random Energy Model in the hypercube in time scales corresponding to the ergodic regime for these dynamics.
We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be co…
We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d \geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are …
Mixed moving average processes appear in the ergodic decomposition of stationary symmetric α-stable (SαS) processes. They correspond to the dissipative part of "deterministic" flows generating SαS processes (Rosinski, 1995). Along these lines we study stationary increment and self-similar SαS processes. Since the classes of stationary increment and self-similar processes can be embedded into the class of stationary processes by the Masani and Lamperti transformations, respectively, we characterize these classes of SαS processes in terms of nonsingular flows and the related cocycles. We illustr…
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying Lévy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the…
We investigate the large deviation behaviour of a point process sequence based on a stationary symmetric stable non-Gaussian discrete-parameter random field using the framework of Hult and Samorodnitsky (2010). Depending on the ergodic theoretic and group theoretic structures of the underlying nonsingular group action, we observe different large deviation behaviours of this point process sequence. We use our results to study the large deviations of various functionals (e.g., partial sum, maxima, etc.) of stationary symmetric stable fields.
We provide asymptotic results and develop high frequency statistical procedures for time-changed Lévy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with respect to the starting point is equal to $\varepsilon$. This setting can be seen as a first step towards a model for tick-by-tick financial data allowing for large jumps. For a wide class of Lévy processes, we introduce a renormalization depending on $\varepsilon$, under which the Lévy process converges in law to an $α$-stable process as $\varepsilon$ goes to $0$…
Let $ξ(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $ξ(k,n)-ξ(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established.
We begin by giving a new proof of the equivalence between the Liouville property and vanishing of the drift for symmetric random walks with finite first moments on finitely generated groups; a result which was first established by Kaimanovich-Vershik and Karlsson-Ledrappier. We then proceed to prove that the product of the Poisson boundary of any countable measured group $(G,μ)$ with any ergodic $(G,\checkμ)$-space is still ergodic, which in particular yields a new proof of weak mixing for the double Poisson boundary of $(G,μ)$ when $μ$ is symmetric. Finally, we characterize the failure of wea…
This paper deals with measurable stationary symmetric stable random fields indexed by R^d and their relationship with the ergodic theory of nonsingular R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish extensions of some structure results of stationary S alpha S processes to SαS fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.
Symmetric Stable Processes on Amenable Groups
本文证明可数可顺从群上平稳非高斯对称α-稳定过程的遍历性等价于弱混合性及Rosinski最小谱表示为null，推广了整数格上的结果并回答了Roy关于幂零群的问题，进而在Heisenberg群等群上构造出弱混合但非强混合的平稳SαS过程。
We establish an abstract local ergodic theorem, under suitable space-time scaling, for the (boundary-driven) symmetric exclusion process on an increasing sequence of balls covering an infinite weighted graph. The proofs are based on 1-block and 2-blocks estimates utilizing the resistance structure of the graph; the moving particle lemma established recently by the author; and discrete harmonic analysis. Our ergodic theorem applies to any infinite weighted graph upon which random walk is strongly recurrent in the sense of Barlow, Delmotte, and Telcs; these include many trees, fractal graphs, an…
We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by $α$-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Harris theorem, and the second on Doeblin's coupling argument. Irreducibility, Lyapunov function techniques, and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite-dimensional equations, introduced in [Priola 2010], with Hölder continuous drift and a general, non-degenerate, sym…
Cutoff ergodicity bounds in Wasserstein distance for a viscous energy shell model with Lévy noise
本文为带有随机能量注入的粘性能量壳格点湍流模型，在重整化WKR距离下建立了显式非渐近遍历界，系统可由多种Lévy过程驱动，基于Bessel函数卷积的解表示证明了渐近急剧热化。
Asymptotic representation theory of general linear groups GL(n,q) over a finite field leads to studying probability measures ρon the group U of all infinite uni-uppertriangular matrices over F_q, with the condition that ρis invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures ρ) was conjectured by Kerov in connection with nonnegative specializations of Hall-Littlewood symmetric functions. Vershik and Kerov also conjectured the following Law of Larg…
We analyze general aspects of exchangeable quantum stochastic processes, as well as some concrete cases relevant for several applications to Quantum Physics and Probability. We establish that there is a one-to-one correspondence between quantum stochastic processes, either preserving or not the identity, and states on free product C*-algebras, unital or not unital respectively, where the exchangeable ones correspond precisely to the symmetric states. We also connect some algebraic properties of exchangeable processes, that is the fact that they satisfy the product state or the block-singleton …
We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclusion process on $\mathbb{Z}$. In this model all particles on sites with one occupied and one empty neighbor jump at each integer time to the empty neighbor site, and if two particles attempt to jump into the same empty site we choose one randomly to succeed. The ETIS states are qualitatively different for densities $ρ<1/2$, $ρ=1/2$, and $1/2<ρ<1$, but in each density region we find states which may be grouped into families, each of which is in natural correspondence with the set of all ergodic measu…
An ergodic theorem with weights and applications to random measures, RW homogenization and IPS hydrodynamics on point processes
本文证明了带加权平均的多维遍历定理，并将其应用于随机测度及点过程上随机游走的淬火均质化，移除了原有技术假设，进而确立了对称简单排他过程淬火流体动力学极限的持续有效性。
On the Ergodicity of Interacting Particle Systems under Number Rigidity
本文建立构形空间上概率测度尾平凡性、输运距离有限性与狄利克雷型不可约性的关联，借助数刚性证明了多类行列式/积和式点过程对应的无穷对数相互作用扩散过程的遍历性。
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