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Let $X$ be a transient Lévy process on $\mathbb R$, and $B\subseteq \mathbb R$ a Borel set with first hitting time $T_B = \inf \left\{t>0 : X_t\in B\right\}$. For Borel $A\subseteq B$, can anything be said about $$ \mathbb P_0(T_A < T_{B\setminus A} | T_A<\infty)? $$ I'm really interested in any results, ideas, even gut feelings. I'd not be surprised if it's related to potential density, but I currently can't see how.

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    $\begingroup$ Would whoever downvoted care to explain why? If there is a problem with my question I'd like to know so that I can avoid it in the future. $\endgroup$
    – user1118
    Commented Nov 8, 2019 at 9:02
  • $\begingroup$ Since we are on R, lets first look at the problem of exiting interval [a,b] from the left endpoint a. For Brownian motion this can be done by studying the exponential martingale and applying optional stopping theorem. See here for the Levy process's exponential martingale math.stackexchange.com/questions/1855860/… $\endgroup$
    – Thomas Kojar
    Commented Nov 14, 2019 at 23:44

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