Filtration of random solutions of a system of linear difference equations with coefficients depending on a Markov chain

We solve the problem of the estimation of a random state for a system with discrete time that is described by a system of linear difference equations with coefficients depending on a finite-valued Markov chain. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 590–592, April, 1998.     

Filtration and prediction of random solutions of a system of linear differential equations with coefficients depending on a finite-valued Markov process

We obtain an equation of optimal filtration for processes of Markov random evolution, which is a solution of systems of linear differential equations with Markov switchings. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 997–1000, July, 1998.     

Small random perturbations of a dynamical system with blow-up

We study small random perturbations by additive white-noise of a spatial discretization of a reaction–diffusion equation with a stable equilibrium and solutions that blow up in finite time. We prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution. For initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study t...     

Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

     

Random periodic solutions of random dynamical systems

In this paper, we give the definition of the random periodic solutions of random dynamical systems. We prove the existence of such periodic solutions for a C¹ perfect cocycle on a cylinder using a random invariant set, the Lyapunov exponents and the pullback of the cocycle.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients

We obtain deterministic first-order linear differential equations with ordinary and variational derivatives and deterministic initial conditions for the expectation and the second moment function of the solution of an ordinary scalar first-order linear inhomogeneous differential equation whose coefficients are random processes. We derive existence conditions for mean periodic solutions. In particular, we consider Gaussian and uniformly distributed random coefficients.     

A system of Markov processes with random lifetimes

Summary Let E be a locally compact Hausdorff space with a countable base, and suppose {x_n} is a countable collection of points in E. Particles enter E at the site x _n according to a Poisson process N _n (t). Upon entrance to E, a typical particle moves through the space, independently of all other particles, according to the transition law of a Markov process, until its death, which occurs at some random time D. We prove several limit theorems for various functional of this infinite particle system. In particular, laws of large numbers, and central limit theorems are proved for occupation times of relatively compact Borel sets.Supported in part by Arizona State University Grant-in-Aid     

Investigation of a system of linear differential equations with random coefficients

We investigate a system of linear differential equations with random coefficients that depend on a periodic Markov process. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1137–1143, August, 1998.     

Optimization of a system of linear differential equations with random coefficients

We consider a system of differential equations with controls that are linearly contained in the right-hand sides. We establish a necessary condition for the optimal control that minimizes a quadratic functional. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 556–561, April, 1999.     

A random attractor for a stochastic second order lattice system with random coupled coefficients

In this paper, we mainly focus on the asymptotic behavior of solutions to the second-order stochastic lattice equations with random coupled coefficients and multiplicative white noises in weighted spaces of infinite sequences. We first transfer stochastic lattice equations into random lattice equations and prove the existence and uniqueness of solutions which generate a random dynamical system. Second we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Then we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero. Finally we present the corresponding results for the system with additive white noises.     

Evolution of a simple dynamical system in a random field

The solution of the Fokker-Planck equation for a nonlinear dynamical system is considered in the framework of perturbation theory. On the basis of uniform estimates, we find a condition for the applicability of the Gaussian approximation of solutions in the neighborhood of an asymptotically stable point of a deterministic system.State University, Bashkir. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 414–419, December, 1993.     

Population dynamical behavior of a single-species nonlinear diffusion system with random perturbation

We consider a single-species stochastic logistic model with the population’s nonlinear diffusion between two patches. We prove the system is stochastically permanent and persistent in mean, and then we obtain sufficient conditions for stationary distribution and extinction. Finally, we illustrate our conclusions through numerical simulation.     

Separable solutions for Markov processes in random environments

In this paper we address the problem of efficiently deriving the steady-state distribution for a continuous time Markov chain (CTMC) S evolving in a random environment E. The process underlying E is also a CTMC. S is called Markov modulated process. Markov modulated processes have been widely studied in literature since they are applicable when an environment influences the behaviour of a system. For instance, this is the case of a wireless link, whose quality may depend on the state of some random factors such as the intensity of the noise in the environment. In this paper we study the class of Markov modulated processes which exhibits separable, product-form stationary distribution. We show that several models that have been proposed in literature can be studied applying the Extended Reversed Compound Agent Theorem (ERCAT), and also new product-forms are derived. We also address the problem of the necessity of ERCAT for product-forms and show a meaningful example of product-form not derivable via ERCAT.     

Stationary availability of a semi‐Markov system with random maintenance

We consider a reparable system with a finite state space, evolving in time according to a semi‐Markov process. The system is stopped for it to be preventively maintained at random times for a random duration. Our aim is to find the preventive maintenance policy that optimizes the stationary availability, whenever it exists. The computation of the stationary availability is based on the fact that the above maintained system evolves according to a semi‐regenerative process. As for the optimization, we observe on numerical examples that it is possible to limit the study to the maintenance actions that begin at deterministic times. We demonstrate this result in a particular case and we study the deterministic maintenance policies in that case. In particular, we show that, if the initial system has an increasing failure rate, the maintenance actions improve the stationary availability if and only if they are not too long on the average, compared to the repairs ( a bound for the mean duration of the maintenance actions is provided). On the contrary, if the initial system has a decreasing failure rate, the maintenance policy lowers the stationary availability. A few other cases are studied. Copyright © 2000 John Wiley & Sons, Ltd.     

Properties of the solutions of a linear system with quasiperiodic coefficients

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 124–129, February, 1990.