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I already asked this question in StackExchange, but found little attention. So I'm just going to copy-paste my original question here.

Let $P$ be a stochastic matrix (of an irreducible Markov Chain) with stationary distribution $\pi^T$ (i.e. $\pi^T P = \pi^T$) and let further $E$ be the matrix of all $1$'s.

Given an $\alpha \in [0,1]$, is it possible to find an expression for the stationary distribution of $$\alpha P + \frac{(1-\alpha)}{n}E,$$ depending on $\pi$ and $\frac{1}{n}\mathbb{1}$, where $\mathbb{1}$ is the vector of all $1$'s?

More generally; given two transition matrices of irreducible Markov Chains $P_1$ and $P_2$ with stationary distributions $\pi_1^T$ and $\pi_2^T$, respectively. Can one find a general formula to calculate the stationary distribution of $$\alpha P_1 + (1-\alpha)P_2 \quad,$$ for $\alpha \in [0,1]$?

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  • $\begingroup$ You seem to be asking, if $P$ and $Q$ are row-stochastic primitive $n \times n$ matrices, then can the normalized left Perron eigenvector of a convex combination of $P$ and $Q$ be reconstructed from those of $P$ and $Q$ (and in the first case, $Q = E$)? (The answer is probably not.) Or are you talking about Markov chains with infinite state space? $\endgroup$
    – David Handelman
    Commented Nov 14, 2017 at 14:42
  • $\begingroup$ Yes exactly, that is what I am asking for. I know the general case might be too ambitious, but for $Q = E$ I had some hope... And no, I am only talking about matrices, i.e. Markov Chains with finite state space. $\endgroup$
    – chickenNinja123
    Commented Nov 14, 2017 at 22:56
  • $\begingroup$ @DavidHandelman Perhaps instead of an explicit computation, there are some useful bounds (in terms of $\alpha$) controlling the total variation distance of the stationary distribution of $\alpha P + (1 - \alpha)Q$ from the stationary distribution of $Q$? $\endgroup$
    – Elle Najt
    Commented Jul 23, 2018 at 19:47

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