Suppose $X$ is a non-explosive diffusion with dynamics

$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$,

where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there any way to formalise this idea? For example, is it possible to define a suitable topology on sample paths of $W$ and construct diffusion sample paths $X(\omega)$ as homeomorphisms of $W(\omega)$?

Part of the motivation for this question comes from the observation that it's possible to do something very similar in the discrete-time case. Given the Euler approximation

$\Delta X_{t+1} = \mu(X_t)\Delta t + \sigma(X_t)\sqrt{\Delta t} W_t $

with $W_t \sim N(0,1)$, then if one knows the values of $\Delta X_t$, then one can unambiguously recover the driving noise $W$. In that sense, one can view $X$ as a transformed version of $W$.