Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

We consider an evolutionary system switched by a semi-Markov process. For this system, we obtain an inhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme.Для систем, що перемикаються иапівмарковськими процесами, одержано результати про неоднорідну дифузійну апроксимацію, де вихідний процес компенсується усередненою функцією в апроксимаційній схемі усереднення

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page

Lifetime distributions in the methods of non-equilibrium statistical operator and superstatistics

A family of non-equilibrium statistical operators is introduced which differ by the system age distribution over which the quasi-equilibrium (relevant) distribution is averaged. To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter - the lifetime of a system. Superstatistics, introduced in works of Beck and Cohen [Physica A \textbf{322}, (2003), 267] as fluctuating quantities of intensive thermodynamical parameters, are obtained from the statistical distribution of lifetime (random time to the system degeneracy) considered as a thermodynamical parameter. It is suggested to set the mixing distribution of the fluctuating parameter in the superstatistics theory in the form of the piecewise continuous functions. The distribution of lifetime in such systems has different form on the different stages of evolution of the system. The account of the past stages of the evolution of a system can have a substantial impact on the non-equilibrium behaviour of the system in a present time moment.Comment: 18 page

Singularly perturbed stochastic systems

Problems of singular perturbation of reducible invertible operators are classified and their applications to the analysis of stochastic Markov systems represented by random evolutions are considered. The phase merging, averaging, and diffusion approximation schemes are discussed for dynamical systems with rapid Markov switchings.Розглянуто проблеми сингулярного збурення оборотних операторів та їх застосування до аналізу стохастичних марковських систем, що задаються випадковими еволюціями. Схеми фазового укрупнення, усереднення та дифузійної апроксимації застосовуються до динамічних систем зі швидкими марковськими переключеннями

Diffusion approximation of stochastic Markov models with persistent regression

Sequences of sums of identically distributed random variables forming a homogeneous Markov chain are approximated by a time-discrete autoregression process of Ornstein-Uhlenbeck type.Послідовність сум незалежних випадкових змінних з однаковим розподілом, що утворюють однорідний марковський ланцюг, апрокснмована авторегресійним процесом з дискретним часом Орнштейна — Уленбека