Local Equivalence of Reversible and General Markov Kinetics

TL;DR

This paper demonstrates that the local velocity cones of general Markov processes and reversible Markov processes with detailed balance are identical at any given distribution, implying their Lyapunov functions coincide.

Contribution

It proves the local equivalence of Markov kinetics and reversible Markov kinetics at the same equilibrium, extending to nonlinear systems with mass action law.

Findings

Velocity cones coincide for all Markov processes and reversible processes at the same P.

Lyapunov functions for reversible and general Markov processes are the same.

Results extend to nonlinear systems with generalized mass action law.

Abstract

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