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 ![[0,1]^N](/images/equations/StationaryDistribution/Inline12.svg)
(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.
