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.

Pontificia Universidad Católica de Chile (PUC-Chile)

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Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

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Santiago – Chile