  Markov process is widely applied in almost all aspects of literature,
especially important for understanding non-equilibrium processes. We introduce
a decomposition to general Markov process in this paper. This decomposition
decomposes the process into 3 independent parts: stationary distribution,
symmetric detailed-balance part and anti-symmetric breaking detailed-balance
part. This complete decomposition captures the steady state as well as the
dynamics of the process, providing an elegant perspective for construction or
analyzing problems. In light of the decomposition, a unique definition of
relative entropy is found to formally separate the effect of detailed-balance
part and breaking detailed-balance part. We find that the relative Gini entropy
production introduced in the paper is not affected by the non-detailed balance
part of the process. This property do not holds for other entropy definition in
general discrete case.
