Limit theorems for semigroups with perturbed generators,with applications to multiscaled random evolutions

For parameters η, let {B(η)} denote infinitesimal operators of strongly continuous semigroups, with resolvents R(λ; B(η)) satisfying λR(λ; B(η)) = P(η) + λV(η) + o(λ). For parameters α, let {A(α)} denote possibly unbounded, linear operators for which {A(α) + B(η)} are infinitesimal operators of strongly continuous semigroups {U_α·η(t)}. For α, η converging simultaneously, we show strong convergence of the semigroups U_α·η(t) to a strongly continuous semigroup U(t), with limiting infinitesimal operator characterized by lim_α·η ∑_jP(η) A(α) × (V(η) A(α))ⁱf. We give applications of the abstract perturbation theorems to limit theorems of random evolutions and associated abstract Cauchy problems, in which multiscaling occurs in the convergence.