Perturbations of Markov processes
Perturbations of Markov processes
复制标题
马尔可夫过程的扰动
DOI：
10.1016/0022-1236(72)90029-8
复制
发表时间：
1972
影响因子：
1.7
通讯作者：
T. Leviatan
中科院分区：
文献类型：
--
作者：
T. Leviatan
考虑了具有可数基的局部紧Hansdorff空间E上的所有实值连续函数的空间C 0（E）上的强连续次Markov预解算子族{V λ：λ> 0}的算子B扰动，使得limx →∞ <$（x）= 0.众所周知，存在一个以{V λ：λ> 0}为预解式族的马尔可夫过程。在B关于{V λ：λ> 0}的“小”条件下，证明了扰动预解族{H λ：λ> λ 0}是C 0（E）上有界算子的强连续预解族.证明了C 0（E）上存在强连续算子半群{St：t <$0}，其预解式为{H λ：λ> λ 0}.还证明了存在一个扩展意义的强马氏过程（Ω F，Ft，θ t Pz），称为具有生成和湮灭的马氏过程，它通过Ex <$（λ t）= Stt <$（x）与扰动半群相联系.给出了该工艺与原工艺的关系。讨论了其在求解偏微分方程柯西问题中的应用。
A perturbation by some operator B, of a strongly continuous sub-Markov resolvent family {V λ: λ> 0} of operators on the space C 0 (E) of all real-valued continuous functions ƒ on a locally compact Hansdorff space E having a countable base, such that lim x→∞ ƒ (x)= 0, is considered. It is well-known that there exists a Markov process having {V λ: λ> 0} as its resolvent family. Under some conditions of “smallness” of B with respect to {V λ: λ> 0} the perturbed resolvent family {H λ: λ> λ 0} is shown to be strongly continuous resolvent family of bounded operators on C 0 (E). It is proved that there exists a strongly continuous semigroup {S t: t⩾ 0} of operators on C 0 (E) having {H λ: λ> λ 0} as its resolvent. It is also proved that there exists a strong Markov process (Ω F, F t, ξ t, θ t P z) of an extended sense called Markov process with creation and annihilation associated to the perturbed semigroup via E x ƒ (ξ t)= S t ƒ (x). Relations are given between this process and the original one. An application to solving the Cauchy problem in partial differential equations is discussed.