Strongly monotone q-functions and a note on strong ergodicity of monotone q-functions

A monotone transition function P(t)P(t) is by definition strongly monotone if it is also a Feller–Reuter–Riley (FRR) transition function. We give a criterion, in terms of q-matrix Q, for existence of strongly monotone Q-function, and show that if exists it is the unique Q-function. As a consequence, we give a criterion for a Feller minimal monotone chain to be strongly ergodic.