We begin with example. For the Poisson process with an intensity $\lambda\_1$ there is an equivalent change of measure which makes it intensity to $\lambda\_2$. 
I would like to find the conditions when is it possible to do with a homogeneous Markov process in a continuous time? It seems to be true for the Markov chains, but it's not true for piecewise deterministic Markov processes (because of pure deterministic part between jumps).


Maybe we can consider at first the diffusion case. Let 
$$
dX\_t = \mu(X\_t)dt+\sigma(X\_t)dw\_t.
$$


Is there a change of measure $P\to Q$ such that under the new measure
$$
dX\_t = a\mu(X\_t)dt+\sigma(X\_t)\sqrt{a}\cdot dw\_t
$$
where $a>0$?



Change of measure Markov process
To address a question asked by the OP in a comment: yes, the LLN yields almost sure results (no expectations here) which show that there exist disjoint sets of complete trajectories such that each Poisson distribution "sees" only one of them (namely $Q\_\lambda$ "sees" only the set $D\_\lambda$ and $Q\_\lambda(D\_\mu)=0$ if $\lambda\ne\mu$). 


About diffusions with different drifts: as you know, if $(W\_t)\_{t\ge0}$ is a Brownian motion under $P$ and if $X\_t=W\_t+\mu t$, then for every $T$, $(X\_t)\_{0\le t\le T}$ is a Brownian motion (without drift) under a measure $Q\_T$ on the space of paths indexed by $[0,T]$. Furthermore, $Q\_T$ is absolutely continuous with respect to the restriction $P\_T$ of $P$ to the space of paths indexed by $[0,T]$ and the density $dQ\_T/dP\_T$ is $Z\_T=\exp(-\mu W\_T-\frac12\mu^2T)$ (this is Cameron-Martin-Girsanov theorem). Thus there exists what you call an equivalent change of measure between $P\_T$ and $Q\_T$. 


But this does not imply that $P$ and $Q$ are mutually absolutely continuous and in fact, they are mutually singular. Note that when $T\to+\infty$, the $P$-martingale $(Z\_T)$ converges almost surely when $T\to+\infty$ to $Z\_\infty=0$... which is not a Radon-Nykodym density. A more direct proof that $P$ and $Q$ are mutually singular is as follows: for every $\mu$, let $L\_\mu$ denote the set of continuous functions $f$ defined on $[0,+\infty[$ such that $f(t)/t\to\mu$ when $t\to+\infty$. If $\mu\ne0$, then $L\_0$ and $L\_\mu$ are disjoint and $P(X\in L\_\mu)=Q(X\in L\_0)=1$.


