I have a question of some integrability of hitting times.

Let $X=(\{X_t\}_{t \ge0},\{P_x\}_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$.

We assume that there exist Borel measurable functions $f\colon E \to [1,\infty)$ and $g \colon E \to \mathbb{R}$ such that $\left\{f(X_t)-f(x)-\int_{0}^{t}g(X_s)\,ds\right\}$ is a local martingale in $P_x$ for any $x \in E$. By convention, we will write $\mathcal{L}f$ for $g$. We further assume that $f$ is bounded, and that there exist $\alpha,\beta \in (0,\infty)$ and compact subset $K \subset E$ such that \begin{align*} \mathcal{L}f(x) \le -\alpha f(x) +\beta\textbf{1}_{K}(x),\quad x \in E. \end{align*}

Under the conditions stated above, we can prove that \begin{align*} \sup_{x \in E}E_x[\sigma_K]<\infty, \end{align*} where $\sigma_K$ is the first hitting time of $K$. Indeed, fix $x \in E \setminus K$ and a localizing sequence $\{\tau_l\}_{l \ge 1}$ . Then, for any $l \ge 1$, $\left\{ f(X_{t\wedge \sigma_K \wedge \tau_l})-f(x)-\int_{0}^{t\wedge \sigma_K \wedge \tau_l}\mathcal{L}f(X_s)\,ds \right\}_{t \ge 0}$ is a $P_x$-martingale. Therefore, we obtain that for any $t>0$ and $l \ge 1$, \begin{align*} -E_{x} \left[\int_{0}^{t \wedge \tau_l \wedge \sigma_K} \mathcal{L}f(X_s)\,ds\right] &= f(x)-E_{x}[f(X_{t \wedge \tau_l \wedge \sigma_K})] \le f(x). \end{align*} Because $\mathcal{L}f \le -\alpha f$ on $E \setminus K$, it follows that \begin{align*} E_{x}[t \wedge \sigma_K \wedge \tau_l] \le f(x)/\alpha. \end{align*} Because $f$ is bounded, Fatou's lemma shows that $\sup_{x \in E \setminus K}E_x[\sigma_K]<\infty$. It clearly holds that $\sup_{x \in K}E_x[\sigma_K]=0.$

**My question**

We can take increasing compact subsets $\{K_n\}_{n=1}^{\infty}$ of $E$ such that $E=\bigcup_{n=1}^{\infty}K_n$. In this situation, I would want to expect that \begin{align*} (1)\quad \lim_{n \to \infty}\sup_{x \in E}E_{x}[\sigma_{K_n}]=0. \end{align*} Then, there exists $N \in \mathbb{N}$ such that for any $n \ge N$, $\sup_{x \in E}E_x[e^{\sigma_{K_n}}]<\infty$.

**Can we prove $(1)$? If not, please tell me a counterexample.**