Column continuous transition functions

A column continuous transition function is by definition a standard transition function P(t) whose every column is continuous for t⩾0 in the norm topology of bounded sequence space l∞. We will prove that it has a stable q-matrix and that there exists a one-to-one relationship between column continuous transition functions and increasing integrated semigroups on l∞. Using the theory of integrated semigroups, we give some necessary and sufficient conditions under which the minimal q-function is column continuous, in terms of its generator (of the Markov semigroup) as well as its q-matrix. Furthermore, we will construct all column continuous Q-functions for a conservative, single-exit and column bounded q-matrix Q. As applications, we find that many interesting continuous-time Markov chains (CTMCs), say Feller–Reuter–Riley processes, monotone processes, birth–death processes and branching processes, etc., have column continuity.