arXiv subjects
Publications and source records attributed to Alexander Veretennikov.
A mean-field extension of the queueing system \(GI/GI/1\) is considered. The process is constructed as a Markov solution of a martingale problem. Uniqueness in distribution is established under a bit different sets of assumptions on intensities.
Polynomial convergence rate to stationarity is shown for extended Erlang -- Sevastyanov's model.
Bounds on convergence rate to the invariant distribution for a class of stochastic differential equations (SDEs) with a gradient-type drift are obtained.
These are lecture notes on the subject defined in the title. As such, they do not pretend to be really new, probably except for the only section about Poisson equations with potentials. Yet, the hope of the author is that they may serve as a bridge to the important area of Poisson equations "in the whole space" and with a parameter, the latter theme not being presented here. Why this area is so important was explained in many papers and books (see the references [12, 34, 35]): it provides one of the main tools in diffusion approximation in the area stochastic averaging. Hence, the aim of these lectures is to prepare the reader to "real" Poisson equations -- i.e., for differential operators instead of difference operators -- and, indeed, to diffusion approximation. Among other presented topics we mention coupling method.
In earlier papers Poisson equation in the whole space was studied for so called ergodic generators $L$ corresponding to homogeneous Markov diffusions ($X_t, \, t\ge 0$) in $\mathbb R^d$. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for PDEs, and secondly with a hope that it may be also useful for some extensions related to homogenization and averaging.
It is be shown that the sequence of Bernstein polynomials for a function of several variables converges to this function uniformly along with every partial derivative of any order, provided that the latter derivative is well defined and continuous. This may be interesting in some questions of stochastic calculus, in particular, in a methodical proof of multidimensional Ito's formula based on the product rule.
A direct approach to linear backward filtering equations for SDE systems is proposed. This preprint is a corrected version of the paper 1995 in the LMS Lecture Notes combined with another paper by the author on the direct approach to linear SPDEs for SDEs.
Recurrence and ergodic properties are established for a single--server queueing system with variable intensities of arrivals and service. Convergence to stationarity is also interpreted in terms of reliability theory.
A new result on stability of an optimal nonlinear filter with respect to small perturbations on every step is established.
A computable estimate of the readiness coefficient for a standard binary-state system is established in the case where both working and repair time distributions possess heavy tails.