Distribution of eigenvalues for the discontinuous boundary-value problem with functional-manypoint conditions

In this study, we investigate the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also a point of discontinuity, a finite number of internal points and abstract linear functionals. So our problem is not a pure boundary-value one. We single out a class of linear functionals and find simple algebraic conditions on the coefficients which guarantee the existence of an infinite number of eigenvalues. Also, the asymptotic formulas for the eigenvalues are found. The results obtained in this paper are new, even in the case of boundary conditions either without internal points or without linear functionals.     

Ergodic decomposition of group actions on rooted trees

We prove a general result about the decomposition into ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. Special attention is paid to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and so-called universal group, are considered in order to demonstrate applications of the theorem.     

On Markov diffusion processes with delayed reflection from boundaries of a segment

A continuous semi-Markov process with values in a closed interval is considered. This process coincides with a Markov diffusion process inside the interval. Thus, violation of the Markov property is only possible at the boundary of the interval. We prove a sufficient condition under which a semi-Markov process is Markov. We show that, in addition to Markov processes with instantaneous reflection from the boundary of the interval. there exists a class of Markov processes with delayed reflection from the boundary. Such a process has a positive average measure of time at which its trajectory belongs to the boundaries. This gives a different proof of a similar result by Gikhman and Skorokhod of 1968. Bibliography: 5 titles.     

Complete asymptotic expansions for lower (upper) continuous random walks with two boundaries. I

The work is devoted to the investigation of some asymptotic properties of probability distributions generated by semicontinuous random walks. Complete asymptotic expansions of the distributions of the boundary functionals connected with two boundaries (a walk in a strip) are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 64–101, 1976.     

On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen’s inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.     

Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries

We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein–Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various computational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.      

Brownian Motion on the Figure Eight

In an interval containing the origin we study a Brownian motion which returns to zero as soon as it reaches the boundary. We determine explicitly its transition probability, prove it is ergodic and calculate the decay rate to equilibrium. It is shown that the process solves the martingale problem for certain asymmetric boundary conditions and can be regarded as a diffusion on an eight shaped domain. In the case the origin is situated at a rationally commensurable distance from the two endpoints of the interval we give the complete characterization of the possibility of collapse of distinct paths.     

On a walk in a strip with inhibitory boundaries

In the paper we have obtained ergodic theorems for walks generated by sums of stationarily connected random variables and two inhibitory boundaries. We have found representations for the steady-state distributions, permitting us to obtain in the case of independent summands a whole series of useful formulas. We discuss the connection of the results established with problems from queueing theory.     

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.     

Markov Chains with Positive Transitions Are Not Determined by Any p-Marginals

 We prove that if is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number p, there exists a continuum of finite valued non Markovian processes which have the p-marginal distributions of X and with positive entropy, whereas for an irrational rotation and essentially bounded real measurable function f with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain. (Received 22 July 1999; in revised form 24 February 2000)     

Markov Chains with Positive Transitions Are Not Determined by Any p -Marginals

 We prove that if is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number p, there exists a continuum of finite valued non Markovian processes which have the p-marginal distributions of X and with positive entropy, whereas for an irrational rotation and essentially bounded real measurable function f with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain.     

Distribution of functionals of Brownian motion stopped at a time that is inverse to local time

Integral functionals of Brownian motion and of Brownian local time, as well as the supremum of Brownian motion and the supremum of Brownian local time are considered. The obtained results allow the computation of the distributions of these functionals for a Brownian motion stopped at the moment when the local time attains first a given value at one of two levels. It has been established that for this stopping time the Brownian local time is a Markov process with respect to the space variable and the generating operator of the process has been found. Examples of the computation of the distributions of certain functionals are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 37–61, 1990.     

On stochastic processes in a multilevel control bulk queueing system

This article analyzes some stochastic processes that arise in a bulk single server queue with continuously operating server, state dependent compound Poisson input flow and general state dependent service process. The authors treat the queueing process as a semi–regenerative process and obtain the invariant probability measure and the transient distribution for the embedded Markov chain. The stationary probability measure for the queueing process with continuous time parameter is found by using semi-regenerative techniques. The authors also study the input and output processes and establish ergodic theorems for some functionals of these processes. The results are obtained in terms of the invariant probability measure for the embedded process and the stationary measure for the queueing process with continuous time parameter     

Law of the iterated logarithm for the periodogram

We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary parts of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and techniques from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.     

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of innnit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.     

The central limit theorem for stationary Markov processes with normal generator—with applications to hypergroups

We extend the central limit theorem for additive functionals of a stationary, ergodic Markov chain with normal transition operator due to Gordin and Lif?ic, 1981 [A remark about a Markov process with normal transition operator, In: Third Vilnius Conference on Probability and Statistics 1, pp. 147–48] to continuous-time Markov processes with normal generators. As examples, we discuss random walks on compact commutative hypergroups as well as certain random walks on non-commutative, compact groups.     

Déviations modérées pour la moyennisation d'une EDS avec petite diffusion

We prove in this Note the moderate deviation principle (MDP) for the averaging principle of a stochastic differential equation (SDE) in a fast random environment, modelized by an exponentially ergodic Markov process independent of the Wiener process driving the SDE. The main tools will be the method of Puhalskii for exponential tightness and a MDP for inhomogeneous functionals of Markov processes established in [5].     

First Passage Time for Brownian Motion and Piecewise Linear Boundaries

We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the first passage time distributions for general nonlinear boundaries. The numerical computation can be easily done by using the Monte Carlo integration, which is straightforward to implement. Some numerical examples are presented for illustration. This approach can be further extended to compute two-sided boundary crossing distributions.     

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.