Take the tour ×  I wanted to know if it was true that if we are
given a one-dimensional Feller process taking values in
$\mathbb{R}$ and a hitting time $\tau_A=\inf\{t>0 s.t. X_t\in
A\}$ with $A$ a open set (this to avoid measurability
complexities but I would be also interested in the general case)
then if $X_0\in \bar{A}^c$ we have :  $\lim_{t\to
0}P[\tau_A<t]=0$   share|improve this question I think it is
more usually called a Feller process –  Henry Feb 24 '11 at
14:53 @Henry : post edited accordingly. –  TheBridge Feb 24 '11
at 18:24 add comment  1 Answer  up vote 3 down vote accepted
Some useful facts about Feller processes are at
http://almostsure.wordpress.com/2010/07/19/properties-of-feller-
processes/.  In particular, every Feller process admits a cadlag
modification. If you are willing to assume you are dealing with
such a modification, then your question becomes pretty easy,
since by continuity of probability $\lim_{t \to 0} P(\tau_A < t)
= P(\tau_A = 0)$. But $\bar{A}^c$ is open and $X_t$ is a.s