There such problem. Let $\xi_t$, $t\in[0,1]$ be a Cauchy process and $\eta_t=\xi_t+Ct$. Find all constants $C\in \mathbb R$ such that distributions of these processes are absolutely continuous wrt each other. It looks like need to show that for any fixed $\{t_1,t_2,..,t_n\}$ for any $\varepsilon>0$ there exist $\delta>0$ such that for any Borel $A$ if $P((\xi_{t_1},..\xi_{t_n})\in A)<\delta$ then $P((\eta_{t_1},..\eta_{t_n})\in A)<\varepsilon$.
If we knew that all finite dimensional distributions of Cauchy process have densities, we would say that the answer is positive for any real $C$, because $\eta$ is made from $\xi_t$ by shift, therefore it has density too.
Is there some information about finite dimensional distribution Cauchy process, or may be there exist another way to solve this problem.