Let $(Z_t)$ be a real continuous Markov process on $[0,1]$ (with regard to its own filtration). Let $f$ be a real mesurable function (not injective).
I would like to know when is $f(Z_t)$ also a Markov process (with regard to the filtration generated by $f(Z_t)$ which is smaller than the original filtration) ?
I looked online and found this paper: https://projecteuclid.org/journals/annals-of-probability/volume-9/issue-4/Markov-Functions/10.1214/aop/1176994363.full
Unfortunately, I am not familiar with the vocabulary of the paper, it is very hard to understand for me. It seems like the answer to my question is in this paper, but I cannot make sense of it. Any help is welcome. What conditions should $f$ and $Z_t$ satisfy?