Let $W$ be a symmetric $\alpha$-stable process with its generator 
$-(-\Delta)^{\alpha/2}$ for some $\alpha \in (0, 2]$ under $\mathbb P$. Let $\mathbb P^x$ be the probability measure induced by a process 
$$X(t) = x + t + W(t)$$ 
starting from $x$, and we set
$$\hat \zeta = \inf\{t>0: X(t) \notin (-1, 1)\}, \quad
\zeta = \inf\{t>0: X(t) \notin [-1, 1]\}.$$
My question is that


[Q.] Is $\mathbb P^x (\hat \zeta = \zeta) = 1$ valid for $\alpha \in (0, 1)$?


[remark]


The answer should be positive if $\alpha \ge 1$. In fact, one can use strong Markov property and time-shift operator to obtain
$$\mathbb P^x(\hat \zeta = \zeta) = 
\mathbb P^x (\zeta \circ \theta\_{\hat \zeta} = 0) = 
\mathbb E^x [ \mathbb P^{X(\hat \zeta)} (\zeta = 0)] = 1.$$
The last equality is true, since $\zeta = 0$ $\mathbb P^{x}$-a.s for all $x\notin (-1, 1)$ when $\alpha\ge 1$. 


However, when $\alpha <1$, $x = -1$ is not regular to $[-1, 1]^c$, i.e. $\zeta>0$ holds almost surely in $\mathbb P^{-1}$. Therefore, the above argument does not work any more, unless $\mathbb P^x(X(\hat \zeta) = -1) = 0$, which is not clear to me.


Thanks.



Exit time of $\alpha$-stable process to an open set and to its closure
If $\alpha < 1$, then $X$ has bounded variation and positive drift, and therefore it does not *creep* downwards; see Theorem 7.11 in Kyprianou's book *Introductory Lectures on Fluctuations of Lévy Processes with Applications*. This means that $$\mathbb{P}^x(X(\tau\_{-1}) \ne -1) = 1,$$ for $x > -1$, where $$\tau\_{-1} = \inf \{t > 0 : X(t) \leqslant -1\}.$$ In particular $$\mathbb{P}^x(X(\hat\zeta) \ne -1) = 1,$$ and consequently $$\mathbb{P}^x(\hat\zeta = \zeta) = 1.$$




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*Edited:* As pointed out by kenneth, the above argument uses $\tau\_{-1}$ defined with the weak inequality, while Theorem 7.11 in Kyprianou's book has a strict inequality, that is,
$$\tau\_{-1^-} = \inf \{t > 0 : X(t) < -1\}.$$
Getting from there to here is not immediate, though!


The proof of Theorem 7.11(i) involves the following reasoning: outside of an event of probability zero, $X(\tau\_{-1^-}) = -1$ if and only if $\inf\_{s \in [0, t]} X(s) = -1$ for some $t > 0$, and the latter event has zero probability due to the fact that the descending ladder-height process has no drift (see Theorem 5.9 in the book).


The same argument actually works for $X(\tau\_{-1}) = -1$, and the proof is in fact slightly simpler. Unfortunately, I do not have time now to write up the details.


