This is a reference request question.

**Statement:** I am interested in an SDE of the form

\begin{equation}\fbox{1}~~~
{\rm d}X_t = f(X_t)\,{\rm d} t + g(t) \, {\rm d} B_t
\end{equation}

Where we assume that $g(t)\in (0,\infty)$ (notice that g(t) is a determistic function), and $f(\cdot)$ satisfies all the conditions required such that $\fbox{1}$ admits a strong continuous solution for all $t\geq 0$.

**Result:** I am interested in a reference of a result of the type $\mathbb{P}( \text{for some }t>0,\,X_t=x)>0$ for all $x \in \mathbb{R}$, in other words $X_t$ can visit any point with some positive probability.

*Remark$^1$: The aforemnetioned conditions imposed on f and g can be changed, the spirit of the question is given an SDE with some necessary regularity conditions the previously described result holds.*

*Remark$^2$: I had posted this question in math.stachexchange and I had not received an answer after an expired bounty and a passage of two weeks. I have now deleted the post on math.stackexchange.*