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We consider continuous--time Markov kinetics with a finite number of statesand a given positive equilibrium distribution P*. For an arbitrary probabilitydistribution $P$ we study the possible right hand sides, dP/dt, of theKolmogorov (master) equations. We describe the cone of possible values of thevelocity, dP/dt, as a function of P and P*. We prove that, surprisingly, thesecones coincide for the class of all Markov processes with equilibrium P* andfor the reversible Markov processes with detailed balance at this equilibrium.Therefore, for an arbitrary probability distribution $P$ and a general systemthere exists a system with detailed balance and the same equilibrium that hasthe same velocity dP/dt at point P. The set of Lyapunov functions for thereversible Markov processes coincides with the set of Lyapunov functions forgeneral Markov kinetics. The results are extended to nonlinear systems with thegeneralized mass action law.

Local Equivalence of Reversible and General Markov Kinetics