Abstract
Let (X, ℬ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X→ X andϕ:X×Y→X, a (small) positive parameterε and a homogeneous Markov chain {y n } in (Y,C) which describes random perturbations. States of the system, say {x ɛn ∈X, n=0, 1,⋯}, are determined by the iteration relations:x ɛn+1 =f(x ɛn )+ɛϕ(x ɛn ,Yn+1) forn≥0, wherex ɛ0 =x 0 is given. Here we study the asymptotic behavior of the solutionx ɛn asε → 0 andn → ∞ under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.
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Supported in part by NSF Grant DMS92-06677.
Supported in part by NSF Grant DMS93-12255.
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Hoppensteadt, F., Salehi, H. & Skorokhod, A. Discrete time semigroup transformations with random perturbations. J Dyn Diff Equat 9, 463–505 (1997). https://doi.org/10.1007/BF02227491
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DOI: https://doi.org/10.1007/BF02227491
Key words
- Difference equations
- random perturbation
- averaging
- diffusion approximation
- randomly perturbed iterations
- stability