Consider simple diffusion $dX\_t = \sigma dw\_t$ and a parameter $a>0$ and $X\_0=x$. Let us denote $Y\_t = X\_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find measure $Q$ such that the process $Y$ is obtained through the process $X$ not with the change of time but with a change of measure, i.e.
$$
\mathcal{Law}(Y\text{ under }P) = \mathcal{Law}(X\text{ under } Q).
$$
Maybe it is possible to find the density process 
$$
Z\_t = \frac{dQ\_t}{dP\_t}
$$
like for the case when we're looking for the martingale measure? Unfortunately, now it seems the Girsanov theorem is useless for my case.



Change of time or change of measure
for example, if $\sigma$ is a constant, then $Y$ satisfies $dY\_t = \sqrt{a} \sigma dW\_t$ so that the law $\mathbb{Q}\_Y$ and $\mathbb{Q}\_X$ of the processes $Y$ and $X$ on the Wiener space $C([0;T],\mathbb{R})$ are generaly singular: in other words $\frac{d \mathbb{Q}\_Y}{d \mathbb{Q}\_X} = 0$ if $|a| \neq 1$.


