Perturbations of Markov processes

AbstractA perturbation by some operator B, of a strongly continuous sub-Markov resolvent family {Vλ : λ > 0} of operators on the space C0(E) of all real-valued continuous functions ƒ on a locally compact Hansdorff space E having a countable base, such that limx→∞ƒ(x) = 0, is considered. It is well-known that there exists a Markov process having {Vλ : λ > 0} as its resolvent family. Under some conditions of “smallness” of B with respect to {Vλ : λ > 0} the perturbed resolvent family {Hλ : λ > λ0} is shown to be strongly continuous resolvent family of bounded operators on C0(E). It is proved that there exists a strongly continuous semigroup {St : t ⩾ 0} of operators on C0(E) having {Hλ : λ > λ0} as its resolvent. It is also proved that there exists a strong Markov process (ΩF,Ft,ξt,θtPz) of an extended sense called Markov process with creation and annihilation associated to the perturbed semigroup via Exƒ(ξt) = Stƒ(x). Relations are given between this process and the original one. An application to solving the Cauchy problem in partial differential equations is discussed