E. Yarovaya

Asymptotics of branching symmetric random walk on the lattice with a single source

Abstract We study the long-time asymptotics of continuous-time branching random walk on ℤd (d ≥ 1) with a single source (i.e., branching site). The random walk is assumed homogeneous, symmetric, irreducible, and having zero mean and finite variance of jumps. We find the limiting extinction probability and the asymptotics of all integer moments for the total population size and for the number of particles at a fixed site.

Prognostic value of changes in arterial stiffness in men with coronary artery disease

Background Men with coronary artery disease (CAD) have been shown to have enhanced arterial stiffness. Arterial function may change over time according to treatment, but the prognostic value of these changes has not been investigated. Objectives The aim of the present study was to assess whether an improvement in large artery rigidity in response to treatment, could predict a more favorable prognosis in a population of men with CAD. Methods A total of 161 men with CAD (mean age 56.8 ± 10.9 years) being treated with conventional therapy underwent brachial-ankle pulse wave velocity (PWVba) measurements at baseline and after six months. Follow-up period was 3.5 years. End-points were major adverse cardiac events (MACE): acute myocardial infarction, unstable angina, coronary intervention, or cardiac death. Results During the three-year follow-up period (since initial six-month follow-up), 30 patients experienced MACE. After six-month follow-up, PWVba had not improved (ΔPWVba ≥ 0%, relative to baseline) in 85 (52.8%) of 161 men (Group 1), whereas it had improved (ΔPWVba < 0%) in the remaining 76 men (47.2%) (Group 2). During follow-up, we noticed 24 cardiovascular events in Group 1 and six events in Group 2 (P < 0.001). Cox proportional hazards analyses demonstrated that independent of conventional risk factor changes, absence of PWVba decrease was a predictor of MACE (RR 3.99; 95% CI:1.81–8.78; P = 0.004). The sensitivity of ΔPWVba was 80% and its specificity was 54%. Conclusions This study demonstrates that an improvement in arterial stiffness may be obtained after six months of conventional therapy and clearly identifies patients who have a more favorable prognosis.

Spectral properties of evolutionary operators in branching random walk models

We introduce a model of continuous-time branching random walk on a finite-dimensional integer lattice with finitely many branching sources of three types and study the spectral properties of the operator describing the evolution of the mean numbers of particles both at an arbitrary source and on the entire lattice. For the leading positive eigenvalue of the operator, we obtain existence conditions determining exponential growth in the number of particles in this model.

Branching Random Walks with Heavy Tails

We consider a continuous-time branching random walk on Z d , where the particles are born and die at a single lattice point (the source of branching). The underlying random walk is assumed to be symmetric. Moreover, corresponding transition rates of the random walk have heavy tails. As a result, the variance of the jumps is infinite, and a random walk may be transient even on low-dimensional lattices (d = 1, 2). Conditions of transience for a random walk on Z d and limit theorems for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained.

Branching Random Walks With Several Sources-super-*

A continuous-time branching random walk on multidimensional lattices with a finite number of branching sources of three types leads to explicit conditions for the exponential growth of the total number of particles. These conditions are expressed in terms of the spectral characteristics of the operator describing the mean number of particles both at an arbitrary point and on the entire lattice.

The structure of the positive discrete spectrum of the evolution operator arising in branching random walks

A branching random walk (BRW) with continuous time and a finite number of branching sources located at points of a multidimensional lattice is considered. The definition of weakly supercritical BRWs, whose discrete spectrum contains a unique positive eigenvalue, is introduced. Conditions for a supercritical BRW to be weakly supercritical are determined.

Supercritical Branching Random Walks with a Single Source

We consider two continuous-time branching random walks on multidimensional lattices with birth and death of particles at the origin. In the first one, the underlying random walk is assumed to be symmetric. In the second one, an additional parameter is introduced to intensify artificially the prevalence of branching or walk at the origin. As a side effect, it violates symmetry of the random walk. Necessary and sufficient conditions for exponential growth for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained. General methods to study the models in the supercritical case are proposed.

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of ‘symmetry’ in the spatial configuration of branching sources. The presented results are based on Green’s function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.

Critical and Subcritical Branching Symmetric Random Walks on d-Dimensional Lattices

We study a symmetric continuous time branching random walk on a d-dimensional lattice with the zero mean and a finite variance of jumps under the assumption that the birth and the death of particles occur at a single lattice point. In the critical and subcritical cases the asymptotic behavior of the survival probability of particles on Z d at time t, as t → ∞, is obtained. Conditional limit theorems for the population size are proved. The models of a branching random walk in a spatially inhomogeneous medium could be applied to the study of the long-time behavior of objects in a catalytic environment.

Multidimensional Watson lemma and its applications

We prove the multidimensional analog of the well-knownWatson lemma and then apply it to prove a local limit theorem for the transition probabilities of symmetric random walks on the multidimensional lattice with infinite variance of jumps.