Dynamical Decomposition of Markov Processes without Detailed Balance

We introduce a dynamical decomposition view in dealing with Markov processes without detailed balance. This work generalizes a previous decomposition framework on continuous-state Markov processes and explicitly gives its correspondence in discrete-state case. We investigate the dynamical roles of decomposed parts by studying the evolution of relative-entropy-like functions. We nnd a special definition of relative entropy to unify the dynamical roles played by the detailed balance part and the breaking detailed balance part. The evolution of the relative entropy na turally bounds the convergence of the process.