Percolation clusters in hyperbolic tessellations

It is known that, for site percolation on the Cayley graph of a co-compact Fuchsian group of genus , infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = P{site is open}. In such cases, the set of limit points at of an infinite cluster is a perfect, nowhere dense set of Lebesgue measure 0. In this paper, a variational formula for the Hausdorff dimension is proved, and used to deduce that is a continuous, strictly increasing function of p that converges to 0 and 1 at the lower and upper boundaries, respectively, of the coexistence phase. Submitted: July 2000.     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

BOREL LINES OF RANDOM DIRICHLET SERIES

For random Diriclilet-haelnacher, Steillhaus and N series of order (R) infinite almostsurely (a.s.), it was proved that a.s. every horizontal line is a Borel line of order (R) illfi-uite and witli a possible exceptional value[6][n. In this paper f by generalized Paley-Zygmundlenru['], sonle sli11ple maPpillgs and Nevanlinlla tlieory[2], we prove that for more general ran-dom Dirichlet random Dirichlet series of order (R) infinite a.s. every horizontal line is a Borelline of order (R) infi…     

全平面上的随机Dirichlet级数的值分布

对于无限级随机Dirichlet级数几乎每一条水平线都是无限级无例外小函数的强Borel线.     

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Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is ??Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is $0s. This result characterizes the slowdown property of the walk.     

Almost sure optimality and optimality in probability for stochastic linear-quadratic regulator with partial information

Partially observed linear-quadratic regulator is considered over an infinite time horizon. A limiting per unit time inequality is proved for the random difference between the cost corresponding to the feedback control based on Kalman filter estimates and the cost corresponding to an alternative control. Under suitable assumptions of admissibility for a control, it is shown that the feedback control mentioned above is asymptotically optimal almost surely and in probability     

The Divisible Sandpile at Critical Density

The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤  1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field.     

Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficients

We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.     

Global divergence of spatial coalescents

We study several fundamental properties of a class of stochastic processes called spatial Λ-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, on any infinite and bounded-degree graph, the total number of particles will always remain infinite at all times, almost surely. Moreover, if ${G\,=\,\mathbb{Z}^d}$ , and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the total number of particles remaining is of order (log* N) ^d at any fixed positive time (where log* is the inverse tower function). At sufficiently large times the total number of particles is of order (log* N) ^d-2, when d?>?2. We provide parallel results in the recurrent case d?=?2. The spatial Beta-coalescents behave similarly, where log log N is replacing log* N.     

Random walk on the infinite cluster of the percolation model

Summary We consider random walk on the infinite cluster of bond percolation on ^d . We show that, in the supercritical regime whend3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell UniversityH.K. was supported in part by the N.S.F. through a grant to Cornell University     

Spectral analysis of percolation Hamiltonians

We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuity-from-the-right argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed Anderson-Quantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard.Mathematics Subject Classification (2000): 35J10,81Q10,82B43     

Lasalle method and general decay stability of stochastic neural networks with mixed delays

This paper investigates the general decay pathwise stability conditions on a class of stochastic neural networks with mixed delays by applying Lasalle method. The mixed time delays comprise both time-varying delays and infinite distributed delays. The contributions are as follows: (1)?we extend the Lasalle-type theorem to cover stochastic differential equations with mixed delays; (2)?based on the stochastic Lasalle theorem and the M-matrix theory, new criteria of general decay stability, which includes the almost surely exponential stability and the almost surely polynomial stability and the partial stability, for neural networks with mixed delays are established. As an application of our results, this paper also considers a two-dimensional delayed stochastic neural networks model.     

Quasi-stationary distributions for structured birth and death processes with mutations

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.     

无限级Dirichlet级数及随机Dirichlet级数

主要研究全平面上无限级Ｄｉｒｉｃｈｌｅｔ级数及随机Ｄｉｒｉｃｈｌｅｔ级数的增长性．对于 Ｄｉｒｉｃｈｌｅｔ级数,研究了它的增长性和正则增长性,得到了它的系数和指数与增长级的两 个充要条件．对于随机Ｄｉｒｉｃｈｌｅｔ级数,证明了它的增长性几乎必然与其在每条水平直线 上的增长性相同．     

无限时滞随机泛函微分方程的Razumikhin型定理

在无限时滞的随机泛函微分方程整体解存在的前提下,建立了一般衰减稳定性的Razumikhin型定理.在此基础上,基于局部Lipschitz条件和多项式增长条件,得到了无限时滞随机泛函微分方程整体解的存在唯一性,以及具有一般衰减速率的p阶矩和几乎必然渐近稳定性定理.     

Nonamenable products are not treeable

LetX andY be infinite graphs such that the automorphism group ofX is nonamenable and the automorphism group ofY has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct productX×Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product has infinitely many connected components almost surely. Research partially supported by NSF grant DMS-9803597.     

Razumikhin-type Theorems on Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory.     

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results.We apply our results to the 2D Navier-Stokes equations forced by a white noise.     

Absence of Infinite Cluster for Critical Bernoulli Percolation on Slabs

We prove that for Bernoulli percolation on a graph , there is no infinite cluster at criticality, almost surely. The proof extends to finite‐range Bernoulli percolation models on ?² that are invariant under ‐rotation and reflection.© 2016 Wiley Periodicals, Inc.     

一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

研究了一类带有限延迟的随机泛函微分方程的Euler-Maruyama(EM)逼近，给出了该方程的带随机步长的EM算法，得到了随机步长的两个特点:首先，有限个步长求和是停时；其次，可列无限多个步长求和是发散的.最终，由离散形式的非负半鞅收敛定理，得到了在系数满足局部Lipschitz条件和单调条件下，带随机步长的EM数值解几乎处处收敛到0.该文拓展了2017年毛学荣关于无延迟的随机微分方程带随机步长EM数值解的结果.