mathematics//probability//stochastic process//Markov chain//stationary distribution
A **stationary distribution** is a distribution over the states of a Markov chain that one step of the chain leaves unchanged, and it is what the chain's long run is read from: the fraction of days a machine spends failed, the share of time a robot spends in each zone, the rank of a web page. For a row-stochastic matrix it satisfies
A stationary distribution is a distribution over the states of a Markov chain that one step of the chain leaves unchanged, and it is what the chain's long run is read from: the fraction of days a machine spends failed, the share of time a robot spends in each zone, the rank of a web page. For a row-stochastic matrix it satisfies
PTπ=π,πi≥0,∑iπi=1.\cb{P}^{\mathsf T}\ca{\pi}=\ca{\pi},\qquad \ca{\pi_i}\ge0,\quad\sum_i\ca{\pi_i}=1.PTπ=π,πi≥0,i∑πi=1.
The distribution in green, the transition matrix in blue. It is a left eigenvector of P\cb{P}P for eigenvalue one, which every stochastic matrix has because its rows sum to one. For the machine that degrades with 2 % a day, fails from degraded with 10 % and is repaired with 50 %, π≈(0.806, 0.161, 0.032)\ca{\pi}\approx(0.806,,0.161,,0.032)π≈(0.806,0.161,0.032): it spends 3.2 % of its days failed, an availability of 96.8 %. The machines keep moving; what is stationary is the proportion of them in each state.
A chain tends to its eigenvector of eigenvalue one, and the second eigenvalue says how fast it forgets where it started.
The distance to π\ca{\pi}π shrinks by a factor ∣λ2∣|\lambda_2|∣λ2∣ per step, the mixing rate. For the machine ∣λ2∣=0.875|\lambda_2|=0.875∣λ2∣=0.875: whatever its state today, the memory of it halves about every five days.
Existence and attraction need different conditions. A finite irreducible chain, where every state reaches every other, has a unique positive stationary distribution; converging to it from any start also needs aperiodicity, return lengths with greatest common divisor one. The swap P=[0110]\cb{P}=\left[\begin{smallmatrix}0&1\1&0\end{smallmatrix}\right]P=[0110] has π=(1/2,1/2)T\ca{\pi}=(1/2,1/2)^{\mathsf T}π=(1/2,1/2)T, yet a distribution on one node alternates forever. A reducible chain can have several closed or absorbing classes, and its end depends on the start. Authors use the word ergodic under competing conventions, so the conditions are stated explicitly.
stationary π(0.50, 0.50) chainSwap Starting in state 1, the distribution evolves by p(k+1) = Pᵀ p(k): p0 = (1.00, 0.00); p1 = (0.00, 1.00); p2 = (1.00, 0.00); p3 = (0.00, 1.00); p4 = (1.00, 0.00). A population of tokens hopping by the same probabilities fluctuates around these numbers.
Press Play and the whole crowd swaps sides at every step, never settling on π; switch on Lazy and the same chain slides onto it.
A walk on a graph with W=WTW=W^{\mathsf T}W=WT and P=D−1W\cb{P}=D^{-1}WP=D−1W has πi=di/∑jdj\ca{\pi_i}=d_i/\sum_jd_jπi=di/∑jdj, checked by πiPij=wij/∑ℓdℓ=πjPji\ca{\pi_i}\cb{P_{ij}}=w_{ij}/\sum_\ell d_\ell=\ca{\pi_j}\cb{P_{ji}}πiPij=wij/∑ℓdℓ=πjPji, detailed balance: high-degree nodes are occupied more, so normalizing transitions removes neither degree effects nor non-uniformity (normalized Laplacian). PageRank is the stationary distribution of such a walk over the web, with a small chance of jumping to a random page so the chain is irreducible.
Metastability is a long stay in a region before escaping, as in two communities connected densely inside and weakly between them; it can be left, unlike an absorbing class (two-community graph). The spectral gap ties mixing to the spectrum, and diffusion and consensus explains why a non-uniform distribution coexists with constant consensus of the dual operator. Theorems in Levin and Peres, Markov Chains and Mixing Times, theorem 4.9.