Yuri Bakhtin
A stochastic differential equation with infinite memory is considered. The drift coefficient of the equation is a nonlinear functional of the past history of the solution. Sufficient conditions for existence and uniqueness of stationary solution are given.
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We continue the study of Markov systems started in \cite{Wer1}. In this paper, we prove a generalization of Breiman's strong low of large numbers \cite{Br} which implies a necessary condition for the uniqueness of the stationary state of a Markov system.
Non-uniqueness of stationary measures for stochastic systems with almost surely invariant manifolds
本文提出一般框架，利用不变子流形的横向Lyapunov不稳定性构造额外平稳测度，证明Lorenz 96模型在阻尼参数足够小时平稳测度发生分岔，产生恰好两个不同的平稳测度。
Let $X$ be an irreducible symmetric Markov process with the strong Feller property. We assume, in addition, that $X$ is explosive and has a tightness property. We then prove the existence and uniqueness of quasi-stationary distributions of $X$.
Uniqueness of stationary compatible probability measures for chains of infinite order with forbidden transitions
本文研究了在可数状态空间上具有禁止转移的无限阶链的 stationary 兼容概率测度的唯一性问题，提出了一组充分条件以保证最多存在一个 stationary 概率测度与核兼容，并扩展了 Johansson 和 Öberg (2003) 提出的 $\ell^2$ 唯一性准则，应用于强非空链，并通过具体例子展示了主要定理及其推论，并与现有文献结果进行了比较。
In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition.
The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's transformation, we also establish the existence result for equations with singular coefficients. Instead of the uniqueness, the non-uniqueness of stationary distributions are considered for equations with regular coefficients. Concrete examples including McKean-Vlasov stochastic equations with the quadratic interaction and the non-quadratic interaction, and equ…
Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: a Banach lattice approach
该研究在弱条件与自然正则性假设下，借助Banach格方法，证明了几乎必然吸收的离散时间马尔可夫链的拟平稳测度和拟遍历测度的存在性与唯一性。
We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits to point out singular phenomenons like non uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non convex environment and requires generalized Laplace's method approximations.
Doubling variables and uniqueness of probability solutions to degenerate stationary Kolmogorov equations
针对具有退化扩散矩阵的平稳Kolmogorov方程，通过直接应用随机分析中的变量加倍方法，得到其概率解唯一性的充分条件。
Stationary distributions of McKean-Vlasov SDEs with jumps: existence, uniqueness, and multiplicity
本文研究带跳McKean-Vlasov随机微分方程平稳分布的存在性、唯一性与多重性，利用Schauder不动点定理和Lyapunov条件等方法，并通过示例展示方程可具有一至三个平稳分布。
We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution.
On the uniqueness of quasi-stationary distributions for population models with spatial structure
本文研究了具有空间结构的人口模型中准稳分布的唯一性，通过引入谱系和分支随机游走，证明了其在添加少量额外信息后具有唯一性。
We present a proof of the almost sure existence, uniqueness and coalescence of directed semi-infinite geodesics in planar growth models that is based on properties of an increment-stationary version of the growth process. The argument is developed in the context of the exponential corner growth model. It uses coupling, planar monotonicity, and properties of the stationary growth process to derive the existence of Busemann functions, which in turn control geodesics. This soft approach is in some situations an alternative to the much-applied 20-year-old arguments of C. Newman and co-authors. Alo…
We study the long time behaviour of a Markov process evolving in $\mathbb{N}$ and conditioned not to hit 0. Assuming that the process comes back quickly from infinity, we prove that the process admits a unique quasi-stationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to infinity). Moreover, we prove that the distribution of the process converges exponentially fast in total variation norm to its quasi-stationary distribution and we provide an explicit rate of convergence. As a first application of our result, we bring a new insight o…
On the Existence and Uniqueness of Stationary Distributions for Some Piecewise Deterministic Markov Processes With State-Dependent Jump Intensity
本文研究一类分段确定性马尔可夫过程，证明其平稳分布与跳跃后位置马尔可夫链的平稳分布一一对应，并据此为随机迭代函数系统情形建立平稳分布存在唯一性判据。
We study the ergodic theory of stationary directed nearest-neighbor polymer models on $\mathbb Z^2$, with i.i.d. weights. Such models are equivalent to specifying a stationary distribution on the space of weights and correctors that satisfy certain consistency conditions. We show that for prescribed weight distribution and corrector mean, there is at most one stationary polymer distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the weights have more than two moments and the corrector mean vector is an extreme point of the superdifferential of the limiting free energy, th…
We consider a Markov process $ X(t) $ on the nonnegative integers $E= S \cup \{0\}$, where $S=\{1,2,...\}$ is an irreducible class and 0 is an absorbing state. In this paper, we investigate conditions under which the quasi-stationary distribution for $X(t)$ exists and is unique, and any initial distribution supported in $S$ is in the domain of attraction of this quasi-stationary distribution. We further find five conditions which are equivalent to that the extinction time is uniformly bounded. As a consequence, we prove the van Doorn's conjecture in \cite{VD2012}. And we can greatly improve th…
In this short paper, we consider a quadruple $(Ω, Å, θ, μ)$,where $Å$ is a $σ$-algebra of subsets of $Ω$, and $θ$ is a measurable bijection from $Ω$ into itself that preserves the measure $μ$. For each $B \in Å$, we consider the measure $μ_B$ obtained by taking cycles (excursions) of iterates of $θ$ from $B$. We then derive a relation for $μ_B$ that involves the forward and backward hitting times of $B$ by the trajectory $(θ^n ω, n \in \Z)$ at a point $ω\in Ω$. Although classical in appearance, its use in obtaining uniqueness of invariant measures of various stochastic models seems to be new. …
Consider the Langevin process, described by a vector (position,momentum) in $\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$ open bounded and connected set of $\mathbb{R}^d$. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain $D:=\mathcal{O}\times\mathbb{R}^d$. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on $D$. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption tow…
An elementary proof of existence and uniqueness of stationary distributions for irreducible Markov chains
该研究针对满足行和为1、元素非负且不可约的n×n矩阵P，利用基础代数和Bolzano-Weierstrass定理，给出了其严格正且和为1的平稳行向量存在唯一性这一经典结果的初等证明。
The present paper studies existence and distributional uniqueness of subclasses of stationary hard-core particle systems arising as thinnings of stationary particle processes. These subclasses are defined by natural maximality criteria. We investigate two specific criteria, one related to the intensity of the hard-core particle process, the other one being a local optimality criterion on the level of realizations. In fact, the criteria are equivalent under suitable moment conditions. We show that stationary hard-core thinnings satisfying such criteria exist and are frequently distributionally …
We show that Zhang's sandpile model (N,[a,b]) on N sites and with uniform additions on [a,b] has a unique stationary measure for all 0 <= a < b <= 1. This generalizes earlier results where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension d >= 1, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure μ. We show that for a stationary ergodic measure μwith density ρ, for all ρ< 1/2, μis stabilizable; for all ρ>= 1, μis not stabilizable; for 1…
We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail $σ$-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to Dobrushin criterion in statistical mechanics. (iii) Results on loss of memory and mixing propert…
We analyze a jump processes $Z$ with a jump measure determined by a "memory" process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.
We study the partial maxima of stationary α-stable processes. We relate their asymptotic behavior to the ergodic theoretical properties of the flow. We observe a sharp change in the asymptotic behavior of the sequence of partial maxima as flow changes from being dissipative to being conservative, and argue that this may indicate a change from a short memory process to a long memory process.
This paper introduces stochastic processes that describe the evolution of systems of particles in which particles immigrate according to a Poisson measure and split according to a self-similar fragmentation. Criteria for existence and absence of stationary distributions are established and uniqueness is proved. Also, convergence rates to the stationary distribution are given. Linear equations which are the deterministic counterparts of fragmentation with immigration processes are next considered. As in the stochastic case, existence and uniqueness of solutions, as well as existence and uniquen…
The random Weierstrass zeta function I. Existence, uniqueness, fluctuations
本文提出一种在给定平稳点过程处具有单位留数简单极点的随机亚纯函数构造方法，刻画了使其减去均值后平稳的有限二阶矩条件，并将其视为 Weierstrass zeta 函数的随机类比或随机点电荷电场。
In this paper, we prove a general result concerning finite-range, attractive interacting particle systems on $\{-1, 1\}^{\mathbb{Z}^d}$. If the particle system has a unique stationary measure and, in a precise sense, relaxes to this stationary measure at an exponential rate then any small perturbation of the dynamics also has a unique stationary measure to which it relaxes at an exponential rate. To augment this result, we study the particular case of Glauber dynamics for the Ising model. We show that for any non-zero external field the dynamics converges to its unique invariant measure at an …
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $α$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with $α$-Fréchet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_1$-topology; in …
In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Levy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart-Pruss-Greiner's theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationar…
In this paper we consider the problem of determining the law of binary stochastic processes from transition kernels depending on the whole past. These kernels are linear in the past values of the process. They are allowed to assume values close to both 0 and 1, preventing the application of usual results on uniqueness. More precisely we give sufficient conditions for uniqueness and non-uniqueness. In the former case a perfect simulation algorithm is also given.
In this paper we aim to find the stationary stochastic viscosity solutions of a parabolic type SPDEs through the infinite horizon backward doubly stochastic differential equations (BDSDEs). For this, we study the existence, uniqueness and regularity of solutions of the corresponding infinite horizon BDSDEs as well as the "perfection procedure" applied to the solutions of BDSDEs. At last the "perfect" stationary stochastic viscosity solutions of SPDEs constructed by solutions of corresponding BDSDEs are obtained.
We explore Ito stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.
Based on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric $α$-stable discrete parameter random field. We show that the power function converges to one as the sample-size increases to infinity under various classes of alternatives having longer memory in the sense of Samorodnitsky(2004). Ergodic theory of nonsingular $\mathbb{Z}^d$-actions play a very important role in the design and analysis of our large sample test.
Well-posedness and stationary distribution of free stochastic differential equations
本文在非交换概率框架下，利用自由伊藤积分，在局部算子Lipschitz和Lyapunov型条件下证明自由布朗运动驱动的自由随机微分方程的整体适定性，并在耗散条件下建立平稳解的存在唯一性。
In this note we consider a chain of $N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov's function technique of Rey-Bellet and coauthors with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by respectively Bernardin and Olla, Lefevere and Schenkel
Long memory or long range dependency is an important phenomenon that may arise in the analysis of time series or spatial data. Most of the definitions of long memory of a stationary process $X=\{X_1, X_2,\cdots,\}$ are based on the second-order properties of the process. The excess entropy of a stationary process is the summation of redundancies which relates to the rate of convergence of the conditional entropy $H(X_n|X_{n-1},\cdots, X_1)$ to the entropy rate. It is proved that the excess entropy is identical to the mutual information between the past and the future when the entropy $H(X_1)$ …
Let Λbe a finite subset of Z^d. We study the following sandpile model on Λ. The height at any given vertex x of Λis a positive real number, and additions are uniformly distributed on some interval [a,b], which is a subset of [0,1]. The threshold value is 1; when the height at a given vertex exceeds 1, it topples, that is, its height is reduced by 1, and the heights of all its neighbours in Λincrease by 1/2d. We first establish that the uniform measure μon the so called "allowed configurations" is invariant under the dynamics. When a < b, we show with coupling ideas that starting from any initi…
Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations
研究随机θ方法生成数值解的平稳分布存在性和唯一性，分析不同θ值对漂移和扩散系数的要求，探讨数值平稳分布收敛性并给出数值实验验证。
We introduce stochastic sequences $ζ(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequences $ζ(k)$ based on their observations at points $ k<0$. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of functionals. Formulas that det…
For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to the predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the fini…
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