On the number of periodic inspections during outbreaks of discretetime stochastic sis. Application to stochastic evolution equations, to appear in. Course notes stats 325 stochastic processes department of statistics university of auckland. Stochastic processes and their applications 43 1992 991 99 northholland almost periodically unitary stochastic processes harry l. Here, we are particularly interested in the study of almost periodicity of those random sequences and their applications to stochastic difference equations. Tudor, stationary and almost periodic solutions of almost periodic affine stochastic differential equations, stochastics stochastics rep.
In particular, the condition of the uniform exponential stability of the linear operator is essentially removed, only using the exponential dichotomy of the linear operator. Squaremean almost periodic solutions of neutral stochastic. Stochastic bifurcation of pathwise random almost periodic. Moreover, the stability analysis on impulsive stochastic di. The topics treated range from existence, uniqueness, boundedness, and stability of solutions, to stochastic difference and differential equations. Renewal processes in most situations, we use the words arrivals and renewals interchangably, but for this type of example, the word arrival is used for the counting process nt. On the basis of this, this article is devoted to the discussion of this problem. Almost periodic solutions and stable solutions for stochastic. Almost periodicity is more likely to accurately describe natural fluctuations diagana et al.
Almost periodically unitary stochastic processes sciencedirect. Finally, we give an example to illustrate the feasibility of our results. Pdf on limit periodicity of discrete time stochastic. It is in part a sequel of authors recent work on almost periodic stochastic difference and differential equations and has the particularity. This book lays the foundations for a theory on almost periodic stochastic processes and their applications to various stochastic differential equations, functional differential equations with delay, partial differential equations, and difference equations. Moreover, the squaremean almost automorphic function is defined. Pseudo almost automorphic in distribution solutions of.
Further, is also mean almost periodic process in uniformly in. The existence of squaremean piecewise almost periodic solutions for linear and nonlinear impulsive stochastic differential equations is established by using the theory of the semigroups of the operators and schauder fixed point theorem. Stm, behavioural science and public health titles, 2012, 30 2, pp. Under a dissipativity condition we prove that the translation of the mean square bounded solution is periodic or almost periodic. For applications in physics and chemistry, see 111. Almost all random variables in this course will take only countably many values, so it is. Essentials of stochastic processes duke university. Squaremean almost periodic solutions nonautonomous. Meansquare almost periodic solutions for impulsive stochastic hostmacroparasite equation on time scales panwang, 1 qingmeilin, 2 andyongkunli 1 department of mathematics, yunnan university, kunming, yunnan, china school of mathematics, wenshan university, wenshan, yunnan, china correspondence should be addressed to yongkun li.
A course on random processes, for students of measuretheoretic. Features 7 this is the first book on almost periodic stochastic processes 7 the topics treated concern a wide audience including mathematicians, statisticians, engineers, and economists 7 advanced graduate students with backgrounds in functional analysis and stochastic analysis may also find the. Under suitable conditions besides lyapunov functions, we obtain the existence of almost periodic solutions in distribution. The concept of the almost automorphic function is proposed by bochner in the paper, which is an important generalization of the classical almost periodic function and is related to some aspects of differential geometry. An extension problem for discretetime almost periodically.
Using the socalled acquistapaceterreni conditions, we establish the existence and uniqueness of a squaremean almost periodic mild solution to those nonautonomous stochastic. These processes are shown to be uniformly almost periodically correlated and hence almost periodically correlated in the sense of gladyshev. Bochneralmost periodicity for stochastic processes. Stochastic bifurcation of pathwise random almost periodic and. Almost periodic stochastic processes pdf free download. Probability and stochastic processes harvard mathematics. The existence and bifurcation of random periodic random almost periodic, random almost automorphic solutions have been. Mbaye, measure theory and s 2pseudo almost periodic and automorphic process. Almost periodic solutions for a class of stochastic. This paper provides sufficient conditions for the sample paths of harmonizable processes to be almost periodic uniformly, stepanov and besicovitch.
Almost none of the theory of stochastic processes cmu statistics. Averaging principle, stochastic reactiondi usion systems, evolution families of measures, almost periodic functions ypartially supported by the nsf grant dms 1407615. B which are squaremean almost periodic is then denoted by ap r. A new criterion ensuring the existence and uniqueness of the quadraticmean asymptotically almost periodic mild solutions for the system is presented. Jul 17, 2006 2007 discrete periodic sampling with jitter and almost periodically correlated processes. By use of the banach fixed point theorem and gronwallbellmans inequality technique on time scales, we obtain the existence and exponential stability of meansquare almost periodic solutions for the hostmacroparasite equation on time scales. Lemma 329 expectations of almostinvariant functions. Bezandry toka diagana almost periodic stochastic processes paul h. Therefore, we must import the stochastic effects into the investigation of impulsive evolution systems.
Pdf almost periodic stochastic processes researchgate. In particular, we established sufficient conditions for convergence of the solution in mean square or almost surely to some stochastic periodic process. Pdf almost periodic stochastic processes is among the few published books that is entirely devoted to almost periodic stochastic processes and their. In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Pdf bochneralmost periodicity for stochastic processes.
The space of the restrictions of all mean almost periodic stochastic processes on is denoted by, and mean almost periodic stochastic processes in, uniformly for in compact subset of by. The concept of poisson almost periodicity is introduced. Stochastic processes i 1 stochastic process a stochastic process is a collection of random variables indexed by time. This is the first book on almost periodic stochastic processes the topics treated. Extending the authors work on almost periodic stochastic difference and differential equations, this book covers almost periodic random processes and applications. A stochastic process is a familyof random variables, xt. Asymptotically almost periodic solutions for a class of. Periodic and almost periodic solutions for semilinear. In this paper, the existence and uniqueness of the squaremean almost periodic solutions to a class of the semilinear stochastic equations is studied.
Stochastic processes sheldon m ross 2nd ed p cm includes bibliographical references and index isbn 0471120626 cloth alk paper 1 stochastic processes i title qa274 r65 1996 5192dc20 printed in the united states of america 10 9 8 7 6 5 4 3 2 9538012 cip. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. The class of harmonizable processes is a natural extension of the class of stationary processes. Asymptotically almost periodic solutions of stochastic. Recently, many authors studied the squaremean piecewise almost periodic solutions for impulsive stochastic evolution equations see 2630.
Almost periodic stochastic processes 9781441994752. Save up to 80% by choosing the etextbook option for isbn. Stochastic processessheldon m ross 2nd ed p cm includes bibliographical references and index isbn 0471120626 cloth alk paper 1 stochastic processes i title qa274 r65 1996 5192dc20 printed in the united states of america 10 9 8 7 6 5 4 3 2 9538012 cip. Almost periodic harmonizable processes springerlink. Meansquare almost periodic solutions for impulsive. Our essential idea is to get stability of solutions or systems by some inherited properties of lyapunov functions. Almost periodic solutions for stochastic differential. The article introduces and studies the concept of pmean almost periodicity for stochastic processes.
Almost periodically unitary stochastic processes core. Squaremean piecewise almost automorphic mild solutions to. Periodic stochastic process rims, kyoto university. In particular, in almost all the considered cases, the two combinations. Existence of almost periodic solutions to some stochastic differential. Hurd associates, raleigh, nc, usa, and centerfor stochastic processes, department of statistics, university of north carolina, chapel hill, nc, usa received 28 february 1990 revised 20 june 1991 a continuous second order complex process xt. Almost all the examples we look at throughout the course can be formulated as markov chains. This chapter deals with discretetime stochastic processes known as random sequences. Introduction this paper is concerned with a class of continuoustime nonstationary second order stochastic processes that are closely related to the almost periodically correlated apc processes introduced by gladyshev and the uniformly almost periodically correlated uapc that have been treated more recently by dehay 8. An alternate view is that it is a probability distribution over a space of paths.
Bochneralmost periodicity for stochastic processes article pdf available in stochastic analysis and applications 302 march 2012 with 147 reads how we measure reads. Periodically and almostperiodically correlated random. Existence and stability of almost periodic solutions to. Covers existence, uniqueness, and stability of solutions for abstract stochastic difference. Research article meansquare almost periodic solutions for. In a deterministic process, there is a xed trajectory. The aim of the special issue stochastic processes with applications is to present a collection.
Almost periodic harmonizable processes swift, randall j. For brownian motion, we refer to 73, 66, for stochastic processes to 17, for stochastic. Preliminaries in this section, b,kk denotes a separable. However, besides impulsive effects, stochastic effects likewise exist in real systems.
We give representations for the processes based on the spectral theory for unitary groups and on the harmonic analysis of uniformly almost periodic functions. That is, at every timet in the set t, a random numberxt is observed. Almost periodic solutions and stable solutions for. Almost periodic and almost automorphic solutions to some. These solutions are pathwise stochastic analogues of deterministic dynamical systems. Motivated by the above discussions, the main aim of this paper is to study the mean square almost periodic solutions for impulsive stochastic di. This work is concerned with the quadraticmean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations. We discuss the problem of the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable hilbert space.
Also, the almost automorphic solutions for differential systems have been extensively investigated in 29. The existence and uniqueness of squaremean almost periodic solutions to some linear and semilinear stochastic differential equations with infinite dimensional levy noise are established provided the coefficients satisfy some suitable conditions. In addition, it establishes a new composition theorem for weighted pseudo almost periodic functions under the non. Mean square almost periodic solutions for impulsive. For brownian motion, we refer to 74, 67, for stochastic processes to 16, for stochastic di. Almost none of the theory of stochastic processes a course on random processes, for students of measuretheoretic probability, with a view to applications in dynamics and statistics cosma rohilla shalizi with aryeh kontorovich version 0. This paper introduces the concept of the piecewise weighted pseudo periodicity for stochastic processes. We treat both discrete and continuous time settings, emphasizing the importance of rightcontinuity of the sample path and. Among others, it will be shown that each pmean asymptotically almost periodic stochastic process is stochastically bounded, and the collection of all pmean asymptotically almost periodic stochastic processes is a banach space when it is equipped with some norm obtained through the norm of l p p, b, where p is the probability measure of the. Under a dissipativity condition we prove that the translation of the mean square. This paper concerns the squaremean almost periodic solutions to a class of nonautonomous stochastic differential equations on a separable real hilbert space. In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by nonautonomous stochastic equations. In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise.
Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such. We investigate the possibility of the periodicity of the solution. Squaremean piecewise almost automorphic mild solutions to a. Bezandry department of mathematics howard university 2441 6th street nw 20059 washington district of columbia usa email protected. Introduction to stochastic processes ut math the university of. Existence of squaremean almost periodic solutions to some stochastic hyperbolic differential equations with infinite delay. In particular, the condition of the uniform exponential stability of the linear operator is essentially removed, only. By developing a single unifying theory, we can easily tackle complex problems with many states and. We consider an impulsive stochastic hostmacroparasite equation on time scales.
Renewal processes since they are arrival processes can be speci. Our abstract results are, subsequently, applied to studying the existence of squaremean almost periodic solutions to some semilinear stochastic equations. Finally, we give an example to illustrate the feasibility of our. This paper introduces the concept of squaremean piecewise almost periodic for impulsive stochastic processes. Existence of almost periodic solutions to some stochastic. Almost periodic stochastic processes is among the few published books that is entirely devoted to almost periodic stochastic processes and their applications. Pdf we compare several notions of almost periodicity for continuous processes.
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