  By killing a stable L\'{e}vy process when it leaves the positive half-line,
or by conditioning it to stay positive, or by conditioning it to hit 0
continuously, we obtain three different positive self-similar Markov processes
which illustrate the three classes described by Lamperti \cite{La}. For each of
these processes, we compute explicitly the infinitesimal generator from which
we deduce the characteristics of the underlying L\'{e}vy process in the
Lamperti representation. The proof of this result bears on the behaviour at
time 0 of stable L\'{e}vy processes before their first passage time across
level 0 which we describe here. As an application, we give the law of the
minimum before an independent exponential time of a certain class of L\'{e}vy
processes. It provides the explicit form of the spacial Wiener-Hopf factor at a
particular point and the value of the ruin probability for this class of
L\'{e}vy processes.
