Markov Processes and Semigroups of Operators

In this paper the relations between various semigroups of operators and between various infinitesimal operators connected with a homogeneous in t Markov process are investigated. General conditions are established under which the Markov process is determined by its corresponding infinitesimal operator. Let Uₜ be a semigroup of linear operators in the Banach space L such that \|Uₜ \| 1. Let Tₜ = Uₜ be an adjoint semigroup in the conjugate space B = L^ *. More abstractly the main object of this paper can be characterized as the study of semigroups Tₜ and its infinitesimal operators in strong and weak topologies of space B.