Bessel process of dimension 2 is defined to be solution of $$ dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0 $$ where $B$ is a standard 1-dimensional Brownian motion. $X$ can be viewed as the norm of a standard 2-dimensional Brownian motion. It returns arbitrarily close to $0$ but will never hit $0$.

It is known that for any $\epsilon>0$, Bessel process of dimension $2-\epsilon$ which is defined to be solution of $$ dX_t=dB_t+\frac{1-\epsilon}{2X_t}dt,\quad X_0=x_0>0 $$ almost surely hits $0$.

Now let $f$ be a certain function such that $f(y)\to 0$ and $f(y)/y\to\infty$ as $y\to 0$. Let $Y_t$ be a solution to $$ dY_t=dB_t+\frac{1-f(Y_t)}{2Y_t}dt. $$ Is there a criterion to judge if $Y_t$ will hit $0$? For example, what will happen when $f(y)=y^c$ for $0<c<1$? What about $f(y)=-y\log y$?