Minimum Power to Maintain a Nonequilibrium Distribution of a Markov Chain

Biological systems use energy to maintain non-equilibrium distributions for long times, e.g. of chemical concentrations or protein conformations. What are the fundamental limits of the power used to "hold" a stochastic system in a desired distribution over states? We study the setting of an uncontrolled Markov chain $Q$ altered into a controlled chain $P$ having a desired stationary distribution. Thermodynamics considerations lead to an appropriately defined Kullback-Leibler (KL) divergence rate $D(P||Q)$ as the cost of control, a setting introduced by Todorov, corresponding to a Markov decision process with mean log loss action cost. The optimal controlled chain $P^*$ minimizes the KL divergence rate $D(\cdot||Q)$ subject to a stationary distribution constraint, and the minimal KL divergence rate lower bounds the power used. While this optimization problem is familiar from the large deviations literature, we offer a novel interpretation as a minimum "holding cost" and compute the minimizer $P^*$ more explicitly than previously available. We state a version of our results for both discrete- and continuous-time Markov chains, and find nice expressions for the important case of a reversible uncontrolled chain $Q$, for a two-state chain, and for birth-and-death processes.