Structure And Dynamical Properties Of Markov Process In A Decomposition View

Markov process is widely applied in physics, chemistry, biology as well as engi-neering modeling. Recently, inspired by biology, researchers developed a decomposi-tion framework in continuous-state Markov process. This framework decomposes theoriginalprocessintothreeparts: potentialfunction,symmetricpartandanti-symmetricpart. In this work, we study and generalize this framework, and fnd its correspondencein discrete-state Markov process. Inspired by this framework, we further obtain somespecial dynamical properties of Markov process.In continuous-state Markov process, we analyze a novel interpretation of stochas-tic diferential equation defned by this framework and derive its relation to classic Itoprocess explicitly. We show that diferent interpretation would lead to dramatic devi-ations by simulation examples and suggest to consider stationary distribution in realworld process modeling.In discrete-state Markov procee, we show the corresponding decomposition tothe continuous one. We further fnd a very special defnition of relative entropy, whosederivative does not depend on the anti-symmetric part. Other defnitions of entropiesdo not have this property. In light of the this special entropy, we further obtain theconvergence eigen value bound. Our formulation is general, and can be universallyapplied.