Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0>0$ is the solution of the SDE $$d X_t = 2\,\sqrt{X_t}\,d W_t+\delta_0\,d t.$$ A naive application of the Girsanov Theorem seems to imply that, in some equivalent new measure (i.e., absolutely continuous w.r.t. the old one), the same process may have a different value $\delta_1$ of the dimension parameter.

I believe that this is not really possible to do rigorously, at least not for an arbitrary pair of values $\delta_0,\delta_1\geq 0$. What is the exact statement of result here --- is it ever possible to change the dimension of a Bessel process by an absolutely continuous change of measure, and if so, under what conditions on $\delta_0$ and $\delta_1$?