Quenched local central limit theorem for random walks in a time-dependent balanced random environment

We prove a quenched local central limit theorem for continuous-time random walks in $${\mathbb {Z}}^d, d\ge 2$$ Z d , d ≥ 2 , in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.

Limit Theorems for a Strongly Supercritical Branching Process with Immigration in Random Environment

We consider a strongly supercritical branching process in random environment with immigration stopped at a distant time 𝑛. The offspring reproduction law in each generation is assumed to be geometric. The process is considered under the condition of its extinction after time 𝑛. Two limit theorems for this process are proved: the first one is for the time interval from 0 till 𝑛, and the second one is for the time interval from 𝑛 till + ∞ +\infty .

Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment

This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 p\geq 1 , and the boundedness of the harmonic moments E ⁢ W n - a \mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log ⁡ Z n \log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.

Mirror descent algorithm on the indefinite control horizon

We consider the problem of optimal control in a random environment in a minimax setting as applied to data processing. It is assumed that the random environment provides two methods of data processing, the effectiveness of which is not known in advance. The goal of the control in this case is to find the optimal strategy for the application of processing methods and to minimize losses. To solve this problem, the mirror descent algorithm is used, including its modifications for batch processing. The use of algorithms for batch processing allows us to get a significant gain in speed due to the parallel processing of batches. In the classical statement, the search for the optimal strategy is considered on a fixed control horizon but this article considers an indefinite control horizon. With an indefinite horizon, the control algorithm cannot use information about the value of the horizon when searching for an optimal strategy. Using numerical modeling, the operation of the mirror descent algorithm and its modifications on an indefinite control horizon is studied and obtained results are presented.

The influence of accidental physical contacts between individuals on viral infection

The paper is devoted to the results of numerical modelling of non-stationary effects during the spread of a viral infection in a small group of individuals. We are considering the case of the spread of a viral infection by airborne droplets. Two consecutive stages of infection of the body are considered. At the first stage, virions enter the lungs and as a result of viremia are transported to the affected organs. In the second stage, the virions actively replicate in the affected organs. Random movement of individuals in the group changes the local concentration of virions near the selected individual. The random level of virion concentration may be greater than a certain critical value after which the infection of the selected individual will go into an irreversible stage. The main purpose of our work is to illustrate qualitatively new effects that occur in nonlinear systems in a random environment.

Resource System with Losses in a Random Environment

The article deals with queueing systems with random resource requirements modeled as bivariate Markov jump processes. One of the process components describes the service system with limited resources. Another component represents a random environment that submits multi-class requests for resources to the service system. If the resource request is lost, then the state of the service system does not change. The change in the state of the environment interacting with the service system depends on whether the resource request has been lost. Thus, unlike in known models, the service system provides feedback to the environment in response to resource requests. By analyzing the properties of the system of integral equations for the stationary distribution of the corresponding random process, we obtain the conditions for the stationary distribution to have a product form. These conditions are expressed in the form of three systems of nonlinear equations. Several special cases are explained in detail.