Comparability and monotonicity of Markov processes

In recent years several papers were published dealing with monotonicity and comparability properties of random, especially Markov processes according to appropriate partial orderings of distribution laws. The present paper comprises a survey about new results in this topic obtained by the authors. In particular, comparability conditions for functionals of comparable processes are given. Moreover comparability and monotonicity properties of Markov processes as well as their functionals are examined and conditions concerning transition rate matrices of time homogeneous Markov processes with discrete state space which guarantee the monotonicity and comparability of the processes are derived.