Christophe Profeta (Université d’Evry Val d’Essonn)

The study of limiting laws, or penalizations, of a given process may
be seen (in some sense) as a way to condition a probability law by an
a.s. infinite random variable. The systematic study of such problems
started in 2006 with a series of papers by Roynette, Vallois and Yor
who looked at Brownian motion perturbed by several examples of
functionals. These works were then generalized to many families of
processes: random walks, Lévy processes, linear diffusions…

We shall present here some examples of penalization of a non-Markov
process, i.e. the integrated Brownian motion, by its first passage
time, nth passage time, and last passage time up to a finite horizon. We shall show that the penalization principle holds in all these cases, but that the
conditioned process does not always behave as expected. Recent results
around persistence of integrated symmetric stable processes will also
be discussed.