The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two non-degenerate $n$ dimensional Brownian motions $W_{1}$ and $W_{2}$ are equivalent to each other.

Intuitively, it may be correct, if we think of these two probability measures by Gaussian measures on the space $C_{0}^{n}(0,1)$ in terms of abstract Wiener space. However, due to the lack of knowledge in this area, I could not find a rigorous proof.

I appreciate if one can provide a detailed proof. Thanks.

In this below, I will given one example to make the question clear: Suppose $W_1$ is 2-D standard BM, $W_2$ is given by $W_2 (t) = [1, 0; \rho, \sqrt{1 - \rho^2}] W_1(t)$ for some $\rho\in (0,1)$. Are $W_1$ and $W_2$ are singular or equivalent?