A stationary distribution of a Markov chain with transition matrix is a vector of probabilities satisfying

If the initial state has statistical distribution , then every later state has the same statistical distribution. Thus is a left eigenvector of with eigenvalue 1, normalized to have nonnegative entries with sum 1.

For a Markov chain on a general state space, a probability measure is stationary when for every measurable set , where is the transition probability from into . The Bak-Sneppen model provides an example on the continuous state space (Alberts et al. 2017).

A finite Markov chain in which every state is reachable from every other has a unique stationary distribution with all entries positive. Convergence to it from every initial statistical distribution additionally requires that the greatest common divisor of the possible state-return times be 1.

For a time-homogeneous continuous-time Markov process on a finite state space, let for and , where is the transition rate from state to state . The stationary-distribution equation becomes

The detailed balance conditions are sufficient for either stationary-distribution equation, but are not necessary.

See also

Detailed Balance, Markov Chain, Markov Process, Minimum Entropy Production Principle, Mixing Time, Probability Vector, Transition Matrix

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References

Alberts, T.; Lee, G. Y.; and Simper, M. "Bak-Sneppen Backwards." Stochastics 89, 1127-1142, 2017. https://doi.org/10.1080/17442508.2017.1282957.Norris, J. R. Markov Chains. Cambridge, England: Cambridge University Press, 1997.

Cite this as:

Weisstein, Eric W. "Stationary Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StationaryDistribution.html

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