I have a question about properties of transient diffusion process.

In the case of $d$-dimensional Brownian motion $B=(B_t,P_x)$ ($d \ge 3$), we can prove that \begin{align} (1)&\quad 0<P_{x}(\sigma_{K}<\infty)\quad \text{ for any }x \in \mathbb{R}^d,\\ (2)&\quad \lim_{|x| \to \infty,\ x \in \mathbb{R}^d}P_{x}(\sigma_{K}<\infty)=0. \end{align} Here, $K$ is a compact subset of $\mathbb{R}^d$ with positive Lebesgue measure and $\sigma_{K}=\inf\{t \ge0 \mid X_t \in K\}$. $|\cdot|$ is the Eucledean metric. Note that (2) follows from the heat kernel estimate of the Brownian motion and the strong Markov property.

**My question**

We consider the next diffusion $X=(X_t,P_x)$ on $\mathbb{R}^d$:
\begin{equation*}
X_t=x+\int_{0}^{t}a(X_s)\,dB_s+\int_{0}^{t}b(X_s)\,ds,
\end{equation*}
where $a$ and $b$ are bounded continuous functions on $\mathbb{R}^d$ and $B$ is the $d$-dim Brownian motion starting from the origin. We assume that $X$ is **transient**.

Is there conditions on $a$ and $b$ such that $\inf_{x \in \mathbb{R}^d}P_{x}(T_{K}<\infty)>0?$

Here, $T_K=\inf\{t\ge0 \mid X_t \in K\}$.