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, $T\_K=\inf\{t\ge0 \mid X\_t \in K\}$.



Hitting probability of a transient diffusion process
I believe that the infimum is always zero. This follows by the following argument, valid for any standard Markov process with infinite lifetime.


Let $T$ be the hitting time of $K$, and write $f(x) = P\_x(T\_K < \infty)$. Suppose, contrary to our claim, that $f(x) \ge c > 0$ for all $x$, and define $g(x) = (f(x) - c) / (1 - c)$.


Fix $x \in \mathbb{R}^d \setminus K$ and $R > |x|$. Let $T\_R$ be the hitting time of $K \cup (\mathbb{R}^d \setminus B(0, R))$ (the first time the process either hits $K$ or exits the centred ball with radius $R$). Then:


1. $T\_R$ is a family of stopping times, increasing in $R$.
2. $T\_R$ converges to $T$, and $T < \infty$ if and only if $T = T\_R$ for $R$ large enough. Indeed, denote by $S$ the limit of $T\_R$. Then $S \le T$, and hence if $S = \infty$, then $T = \infty$. Let us see what happens if $S < \infty$. By quasi-left continuity, the finite limit $\lim\_{R \to \infty} X\_{T\_R}$ exists and it is equal to $X\_S$. Since $X\_{T\_R}$ either belongs to $K$ or to $\mathbb{R}^d \setminus B(0, R)$, convergence of $X\_{T\_R}$ to a finite limit implies that $X\_{T\_R} \in K$ for all $R$ large enough. In particular, $T = T\_R$ for large enough, and consequently $S = T = T\_R$ and $X\_T = X\_{T\_R} \in K$ for $R$ large enough.
3. By the strong Markov property, $$f(x) = P\_x(T < \infty) = E\_x(f(X\_{T\_R})) , $$
so similarly $g(x) = E\_x(g(X\_{T\_R}))$ (here we use infinite lifetime: constants are harmonic).


Since $g$ is non-negative and equal to $1$ on $K$, we have
$$ P\_x(T = T\_R) = P\_x(X\_{T\_R} \in K) \le E\_x(g(X\_{T\_R})) = g(x) $$
for $R$ large enough. Passing to the limit as $R \to \infty$, we find that
$$ f(x) = P\_x(T < \infty) = \lim\_{R \to \infty} P\_x(T = T\_R) \le g(x) , $$
contrary to the definition of $g$.


(I may be getting something slightly wrong due to the usual problem whether the hitting time is defined with $t > 0$ or $t \ge 0$ in the infimum, but this should be minor).


