The Poincaré Map of Randomly Perturbed Periodic Motion

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.     

Property of “correct exit” and one limit theorem for semi-Markov processes

A necessary and sufficient condition on weak convergence of a sequence or probability measures in the space D_{[o, ∞]}(X) is formulated in terms of first exit times. The proof of necessity is based on continuity of first exit times and first exit points with respect to the Stone-Skorokhod metric on the set of functions that “correctly exit” from an open set Δ?X. A limit theorem for semi-Markov processes is proved as an application.     

EXIT DISTRIBUTION LEAVING A BALL FROM THE CENTER AND BROWNIAN MOTION

It is skowed that if the first exit distribution leaving any ball from the center is theuniform distribution on the sphere, then the Levy process is a scaled Brownian motion.The paper also gives a characterization of a continuous Hunt process by the first exitdistribution from any ball.     

On Several Two-Boundary Problems for a Particular Class of Lévy Processes

Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms.     

Level crossing problems and drift reliability

Reliability of products is often determined by special technical or physical drift parameters. Suitable stochastic processes are applied to model such drift processes. An failure can be observed when the drift process leaves a given region at the first time. Then the lifetime is the random time to first exit. Applying the first exit theory it is possible to find the type of lifetime distribution. Further the parameters of the lifetime distribution are to be estimated by observations of realisations of the underlying drift process.     

Asymptotics for a Curve with a Given Distribution Density of the First Exit Position for a Wiener Process

The following inverse first exit problem for a Wiener process is considered: to find an upper class with a given distribution of the first exit point from the domain bounded by this curve. Some estimates are obtained for a curve with a given density at zero. Bibliography: 2 titles.     

Decision Making Times in Mean-Field Dynamic Ising Model

We consider a dynamic mean-field ferromagnetic model in the low-temperature regime in the neighborhood of the zero magnetization state. We study the random time it takes for the system to make a decision, i.e., to exit the neighborhood of the unstable equilibrium and approach one of the two stable equilibrium points. We prove a limit theorem for the distribution of this random time in the thermodynamic limit.     

Law of two-sided exit by a spectrally positive strictly stable process

For a spectrally positive strictly stable process with index in (1, 2), we obtain (i) the sub-probability density of its first exit time from an interval by hitting the interval’s lower end before jumping over its upper end, and (ii) the joint distribution of the time, undershoot, and jump of the process when it makes the first exit the other way around. The density of the exit time is expressed in terms of the roots of a Mittag-Leffler function. Some theoretical applications of the results are given.     

Scaling limit for the diffusion exit problem in the Levinson case

The exit problem for small perturbations of a dynamical system in a domain is considered. It is assumed that the unperturbed dynamical system and the domain satisfy the Levinson conditions. We assume that the random perturbation affects the driving vector field and the initial condition, and each of the components of the perturbation follows a scaling limit. We derive the joint scaling limit for the random exit time and exit point. We use this result to study the asymptotics of the exit time for 1D diffusions conditioned on rare events.     

左截尾双参数指数分布的可靠寿命的广义置信下限

本文基于左截尾双参数指数分布定数截尾数据,利用Weerahandi给出的广义枢轴量和广义置信区间的概念,通过两种不同的方法建立了可靠寿命的广义置信下限.第1种方法利用位置参数无限制时可靠寿命的广义置信下限来定义左截尾情形下可靠寿命的限制广义置信下限,第2种方法基于广义枢轴量在限制参数空间上的条件分布给出可靠寿命的条件广义置信下限.我们分别研究了这两种置信下限的性质,给出了简单易行的数值计算方法.模拟比较表明限制广义置信下限具有好的覆盖率性质,条件广义置信下限的覆盖率与参数取值有关,但它有时比限制广义置信下限具有更大均值和更小标准差.     

The first exit time and ruin time for a risk process with reserve-dependent income

This paper investigates the first exit time and the ruin time of a risk reserve process with reserve-dependent income under the assumption that the claims arrive as a Poisson process. We show that the Laplace transform of the distribution of the first exit time from an interval satisfies an integro-differential equation. The exact solution for the classical model and for the Embrechts–Schmidli model are derived.     

Exit problems associated with finite reflection groups

We obtain a formula for the distribution of the first exit time of Brownian motion from a fundamental region associated with a finite reflection group. In the type A case it is closely related to a formula of de Bruijn and the exit probability is expressed as a Pfaffian. Our formula yields a generalisation of de Bruijn’s. We derive large and small time asymptotics, and formulas for expected first exit times. The results extend to other Markov processes. By considering discrete random walks in the type A case we recover known formulas for the number of standard Young tableaux with bounded height.Mathematics Subject Classification (2000): 20F55, 60J65     

Filtering partially observable diffusions up to the exit time from a domain

We consider a two-component diffusion process with the second component treated as the observations of the first one. The observations are available only until the first exit time of the first component from a fixed domain. We derive filtering equations for an unnormalized conditional distribution of the first component before it hits the boundary and give a formula for the conditional distribution of the first component at the first time it hits the boundary.     

Two-boundary problems for a Poisson process with exponentially distributed component

For a Poisson process with exponentially distributed negative component, we obtain integral transforms of the joint distribution of the time of the first exit from an interval and the value of the jump over the boundary at exit time and the joint distribution of the time of the first hit of the interval and the value of the process at this time. On the exponentially distributed time interval, we obtain distributions of the total sojourn time of the process in the interval, the joint distribution of the supremum, infimum, and value of the process, the joint distribution of the number of upward and downward crossings of the interval, and generators of the joint distribution of the number of hits of the interval and the number of jumps over the interval. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 922–953, July, 2006.     

A deterministic‐control‐based approach motion by curvature

The level‐set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two‐person game. More precisely, we give a family of discrete‐time, two‐person games whose value functions converge in the continuous‐time limit to the solution of the motion‐by‐curvature PDE. For a convex domain, the boundary's “first arrival time” solves a degenerate elliptic equation; this corresponds, in our game‐theoretic setting, to a minimum‐exit‐time problem. For a nonconvex domain the two‐person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the “positive part of the curvature.” These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first‐order Hamilton‐Jacobi equation. Our situation is different because the usual first‐order calculation is singular. © 2005 Wiley Periodicals, Inc.     

Comparison theorems for exit times

We study bounds on the exit time of Brownian motion from a set in terms of its size and shape, and the relation of such bounds with isoperimetric inequalities. The first result is an upper bound for the distribution function of the exit time from a subset of a sphere or hyperbolic space of constant curvature in terms of the exit time from a disc of the same volume. This amounts to a rearrangement inequality for the Dirichlet heat kernel. To connect this inequality with the classical isoperimetric inequality, we derive a formula for the perimeter of a set in terms of the heat flow over the boundary. An auxiliary result generalizes Riesz' rearrangement inequality to multiple integrals. Submitted: February 2000, Revised version: December 2000, Final version: May 2001.     

Two-boundary problems for a random walk

We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed jumps. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1485–1509, November, 2007.     

Asymptotics of random convex polygonal lines

The present paper deals with the limit shape of random plane convex polygonal lines whose edges are independent and identically distributed, with finite first moment. The smoothness of a limit curve depends on some properties of the distribution. The limit curve is determined by the projection of the distribution to the unit circle. This correspondence between limit curves and measures on the unit circle is proved to be a bijection. The emphasis is on limit distributions of deviations of random polygonal lines from a limit curve. Normed differences of Minkowski support functions converge to a Gaussian limit process. The covariance of this process can be found in terms of the initial distribution. In the case of uniform distribution on the unit circle, a formula for the covariance is found. The main result is that a.s. sample functions of the limit process have continuous first derivative satisfying the Hölder condition of order a, for any fixed a with 0相似文献   

高维Kramers系统离出点的分布问题

通过对高维Kramers系统与之对应稳态Fokker-Planck方程的渐近分析,仔细探讨了该系统在平衡点吸引域的边界上离出点的分布问题.运用变量替换、匹配原理、局部坐标变换、边界层展开等方法,对外解、远离鞍点处的边界层及鞍点处的边界层进行分析,得出离出点分布的渐近表达式.