I am dealing with a topological question concerning the mapping that maps a positive stochastic matrix onto its invariant distribution. I am asking myself if such a mapping is continuous (or measurable) with respect to suitable topologies ($\sigma$-fields). Let me describe the problem some more precisely.

Let $n>1$ and $S(n,+)$ be the space of (element-wise) positive (row-)stochastic matrices and let $A\in S(n,+)$.

Due to the Perron-Frobenius theorem, $\rho(A)=$1 is the unique eigenvalue on the spectral radius of $A$ with corresponding (right) eigenvector $v=v_A>0$, where $v$ is normalized, i.e. it is (row-)stochastic and hence can be viewed to be the unique $A$-invariant distribution. Furthermore, $B:=B_A:=\lim\limits_{n\rightarrow \infty}A^n=\lim\limits_{n\rightarrow \infty}\frac{1}{n}\sum\limits_{k=1}^n A^k\in S(n,+)$, where the rows of $B$ are all equal and are all stochastic (left) eigenvectors of $A$ and $BA=AB=B$ as well as $B^2=B$. Thus, $B$ is a projection, called the Perron-projection and can be written as $B=vw^t$, where $v,w$ are such that $Av=v$, $w^tA=w^t$ and $w^tv=1$.

I am wondering if the mapping $A\mapsto B_A$ or $A\mapsto v_A$, respectively, is continuous with respect to the topologies of coordinate-wise convergence.

In other words, do minor changes in the positive stochastic matrix only give rise to minor changes in the corresponding one-dimensional Perron-eigenspaces?

How can this problem best be regarded and approached? In particular in situations where it is not efficient to solve the identity $w^t A=w^t$ explicitly.

If it was a known problem, please let me know where it is described or solved, respectively.

Thanks in advance and regards