column: $w_{ik}$. ](transition3.eps){width="8cm"}

Important features of the master equation (\[MasterEqn\]) are:

1\) The probability vector ${\bm f}_{t}$ should be positive and normalizable. Let ${\bm 1} = (1,1,\dots 1)$ be a $K$-dimensional vector, then we set $ {\bm 1}\cdot {\bm f}_{t} =1.$ More important, because of normalization, $\sum_{k=1}^Kw_{ik} =1$, one has $ \hat W \cdot {\bm 1}= {\bm 1}$. This implies that the probability is conserved under the evolution ${\bm 1}\cdot {\bm f}_{t+1} =  {\bm 1}\cdot \hat W^\dagger {\bm f}_{t}= {\bm 1}\cdot {\bm f}_{t} = 1.$

2\) The Perron-Frobenius equation maybe solved exactly, provided an initial given distribution ${\bm f}_0$: ${\bm f}_{t} = (\hat W^\dagger )^t {\bm f}_{0}$.

3\) Because of the Frobenius theorem, it exists an eigenvalue which is one, $\lambda_1=1$, while others eigenvalues are inside the unitary circle $|\lambda_i|<1$ for $i>1$. Let ${\bm f}_{eq} $ be the Eigenvector associated with the Eigenvalue $\lambda_1=1$; this is an invariant vector ${\bm f}_{eq} = \hat W^\dagger  {\bm f}_{eq} $.

4\) In the following, we denote by ${\bm \chi}^{(i)}$ the eigenvectors of $\hat W^\dagger$ corresponding to $\lambda_i$. Naturally one has ${\bm \chi}^{(1) }\equiv  {\bm f}_{eq}.$

5\) The existence of an equilibrium state: $\lim_{t\to\infty} {\bm f}_{t} = {\bm f}_{eq}$.

6\) Because all elements in 