I'm interested in the following question:
Given a parameter $t\in \mathbb{R}$ and a column stochastic matrix $P(t)$ (i.e., $e^T P(t)=e^T$ and $P(t)_{ij}\ge 0$), calculate the Lipschitz constant of the (unique and positive) right eigenvector $\pi$ of eigenvalue 1, such that $\|\pi\|_1=1$, under variations in $t$.
This is, if $P(t_1)\pi_1=\pi_1$ and $P(t_2)\pi_2=\pi_2$ for any $t_1$ and $t_2$, obtain a constant $L$ such that $\|\pi_1-\pi_2\|\le L\|t_1-t_2\|$.
It is known that the constant $L$ is related to the derivative $\frac{d \pi}{dt}$, and thus to the following question:
Response of stationary distribution to perturbation of a stochastic matrix
However, this question does not tackle the case of $\sum_i \pi(i)=\|\pi\|_1=1$, that is the case that interests me. It would also be nice if the second derivative could also be formalized. Could you help me?
Thank you very much in advance.