The minimal KL divergence rate D(P* || Q) lower-bounds the power to control a Markov chain to a desired stationary distribution, with explicit solutions for reversible chains and birth-death processes.
From level 2.5 to level 2 large deviations for continuous time Markov chains,
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Minimum Power to Maintain a Nonequilibrium Distribution of a Markov Chain
The minimal KL divergence rate D(P* || Q) lower-bounds the power to control a Markov chain to a desired stationary distribution, with explicit solutions for reversible chains and birth-death processes.