I feel it always necessitates a certain amount of work before reaching the conclusion that the transform of a diffusion by a function is not a Markovian diffusion.
I was wondering if there were known necessary conditions over the transform (and/or maybe the diffusion process), that should be satisfied for the transform to stay a Markovian diffusion.
Hopefully such conditions would give a "methodology" (or even better an explicit calculus) to discard some transforms to be Markov diffusion processes.
Let's formalize the problem (or a simplified version of it). So given a Itô-diffusion (let's stay one-dimensional for the moment) $X_t$ obeying an SDE such as :
$dX_t=b(X_t)dt+a(X_t)dW_t$ (with enough regularity so that $X_t$ exists, and $W_t$ is 1-dim Brownian Motion)
And a function $f$, what are the conditions over $f$ so that $f(X_t)$ stays a diffusion, i.e. can be written as :
Reference or proofs are welcome and so is extension to multivariate case.