Consider a canonical Wiener space $(\Omega,\mathscr{F},\mathbb{P})$. Consider two SDEs:
$$dX_t = \alpha(X_t,t)dt+dW_t, \ \ \text{Law}(X_0)=\mu$$ and $$dY_t = \beta(Y_t,t)dt+dW_t, \ \ \text{Law}(Y_0)=\nu$$
Suppose both SDES have solutions (say unique strong solutions). Is there a way to use Girsanov to compare the laws of these two SDEs? I am interested in both the path measures and the time marginals.
This cannot be done in general, since if $\mu \perp \nu$, (say, $X_0=x,Y_0=y$ with $y\neq x$) the two path measures must be singular. If $\mu=\nu$, the usual Girsanov change of measure does the job: if $\mathscr{E}(M)$ is the exponential martingale operator, we can use Girsanov to define a probability measure $\mathbb{Q}$ with density wrt to $\mathbb{P}$ such that
$$\frac{d\mathbb{Q}}{d\mathbb{P}}=\mathscr{E}\left( \int_0^T -\alpha_y(Y_t)+\beta_t(Y_t)dt\right)=\frac{\text{Law}(Y_{[0,T]})}{\text{Law}(X_{[0,T]})}$$
I would like to know if something similar can be done if $\mu \neq \nu$, but under the assumption that $\nu \sim \mu$.