@ David: Did you consider the implications of Girsanov theorem for this setting? Let $\alpha_t = \mu((W_s)_{0 \leq s \leq t}$. By a change of measure of the form $\exp\left(\int_0^t \alpha_s d W_s - 1/2 \int_0^t \alpha_s^2 \ ds \right)$ (perhaps with different sign before the Ito integral) your process $W$ transforms into a Brownian motion. Girsanov theorem is classical and can be found everywhere e.g. in Oksendal or Karatzas/Shreve.