Let us consider a Markov chain with transition probability $P$, such that this Markov chain is irreducible and aperiodic, with stationary distribution $\pi$. Now, let us consider another transition probability $Q$ on the same state space, such that the resulting Markov chain is also irreducible and aperiodic, with stationary distribution $\mu$. We start from the same initial state distribution $d$. Assuming that $P$ and $Q$ are close to each other in some sense, is there anything we can say about the distance between $\pi$ and $\mu$, for example in total variation?
My initial take was to use a triangle inequality on that distance to write something like (informally):
$$ ||\pi - \mu|| \leq ||\pi - P^td|| + ||P^td - Q^td|| + ||Q^td - \mu|| $$
for any $t$. The first and the last term can be bounded using a convergence theorem, but I can't really see anything interesting to say about the second term.
Thanks!