Discrete Time Semigroup Transformations with Random Perturbations

Let (X, ℬ)) and ( Y, ℓ) be two measurable spaces with X being a linear space. A system is determined by two functions f(X): X → X and φ: X × Y → X, a (small) positive parameter ε and a homogeneous Markov chain {yn} in ( Y, ℓ) which describes random perturbations. States of the system, say {xnε ∈ X, n = 0, I ,...}, are determined by the iteration relations: xn + 1c = f(xnε) + εφ(xnε, yn + 1) for n ≥ 0, where x0ε = x0 is given. Here we study the asymptotic behavior of the solution xnε as s → 0 and n → ∞ under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics. © 1997 Plenum Publishing Corporation.