Let $X_t$ and $Y_t$ be $d$-dimensional Itô diffusion processes that solve following SDEs,
$\mathrm{d}X_t = \alpha X_t \mathrm{d}t + \Sigma \mathrm{d}B_t\,$ where $\,B_t$ is a standard brownian motion, $\,\Sigma \Sigma^\top$= a positive semi-definite (Cov matrix), and $(\Sigma \Sigma^T)_{ii} = \sigma^2$.
$\mathrm{d}Y_t = \beta Y_t \mathrm{d}t + \sigma I_d \mathrm{d}B_t$.
Here $X_0$ and $Y_0$ follow some distributions. $\alpha, \beta \in R$, $\sigma > 0$ (scalars), and $I_d$ is the $d$-dimensional identity matrix. Let $\mathbb{P}$ and $\mathbb{Q}$ be measures associated by the processes $X_t$ and $Y_t$ respectively.
I'm wondering if $\mathbb{P}$ and $\mathbb{Q}$ are mutually singular? (does $\mathrm{d}\mathbb{P}/\mathrm{d}\mathbb{Q}$ exist?) If they are singular, how can I show it?
I'm guessing $i$-th element of $X_t$, denoted by $X^{(i)}_t$, follows $\textrm{d} X_t^{(i)} = \alpha X_t^{(i)} \textrm{d}t + \sigma \mathrm{d} B_t^{(i)}$ (Doesn't it?). So I assume that $\mathbb{P}$ and ${\mathbb{Q}}$ won't be mutually singular. Since I don't have a strong background on probability theory, I'm looking for some references that help me drawing some conclusions. I would appreciate any pointers and hints :)