A Reduction Process for Perturbed Markov Chains

Given a perturbed Markov chain with generator A( ε ) = A₀ + ε A₁ + ε ² A₂ + ⋯ it is shown that to each time scale ( t,t /ε , t/ε ² , ⋯ ) is associated a reduced Markov chain. This process of reduction has only finitely many steps. It is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage. It is shown that if λ is an eigenvalue (different from zero) of the generator of the kth reduced chain with multiplicity p then A( ε ) admits p eigenvalues λ ( ε ) such that λ ( ε ) = ε ᵏ λ + o( ε ᵏ ). A basis of A₀ (resp. A₀^ *) which is robust to the perturbation is constructed.