On the absolute continuity of measures corresponding to processes of diffusion type relative to a Wiener measure

In this work there are given necessary and sufficient conditions for the absolute continuity and equivalence ($\mu_\xi\ll\mu_\omega$, $\mu_\omega\ll\mu_\xi$, $\mu_\xi\sim\mu_\omega$) of a Wiener measure $\mu_\omega$ and a measure $\mu_\xi$ corresponding to a process $\xi$ of diffusion type with differential $d\xi_t=a_t(\xi)\,dt+d\omega_t$.
The densities (the Radon–Nikodým derivatives) of one measure with respect to the other are found. Questions of the absolute continuity and equivalence of measures $\mu_\xi$ and $\mu_\omega$ are investigated for the case when $\xi$ is an Ito process. Conditions under which an Ito process is of diffusion type are derived. It is proved that (up to equivalence) every process $\xi$ for which $\mu_\xi\sim\mu_\omega$ is a process of diffusion type.