Conditions for the absolute continuity of two diffusions

Abstract:

Consider two diffusion processes on the line. For each starting point x and each finite time t, consider the measures these processes induce in the space of continuous functions on [0, t]. Necessary and sufficient conditions on the generators are found for the induced measures to be mutually absolutely continuous for each x and t. If the first process is Brownian motion, the second one must be Brownian motion with drift $b(x)$, where $b(x)$ is locally in ${L_2}$ and satisfies a certain growth condition at $\pm \infty$.