Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \mathbb{R}^n$ such that $$ \phi(X_t)= Y_t? $$

More generally, since the continuous image (in the above sense) of a diffusion process needs not be a diffusion process since the Ito-formula (and extensions thereof) fail. What conditions do we need on $X,Y$ and $\phi$ such that $Y_t$ is just an $\mathbb{R}^n$-valued Markov process?