Nicolas Juillet, Université de Haute-Alsace (UHA), Mulhouse: Markovinification of the quantile process (Oberseminar Mathematische Stochastik)
Wednesday, 08.12.2021 17:00 im Raum SRZ 216
It has been established by Lisini that absolutely continuous curves (of order 2) $t\to \mu_t$ in the Wasserstein space over a metric space $X$ can be represented by an action-minimizing probability measure on the space of absolutely continuous curves. We will show that in the basic case of the real line ($X=\R$), this measure can moreover be asked to be Markovian. This is a special case of a more general result, with consequences in stochastics, where no continuity assumptions are made on the family $\mu$. (joint work with Charles Boubel from Strasbourg)
Julien Poisat, Univ. Paris-Dauphine: Simple random walk among power-law renewal obstacles (Oberseminar Mathematische Stochastik)
Wednesday, 15.12.2021 17:00 im Raum SRZ 216
We consider a one-dimensional simple random walk among static soft obstacles. The walk has a certain fixed positive probability to be killed each time it meets one of these obstacles. The positions of the obstacles are sampled independently from the walk and according to a renewal process. The distribution of the gaps between consecutive obstacles is assumed to have a power-law decaying tail. First, we prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. Then, we prove a localization property for the random walk conditioned to survive for a long time, in the spirit of one-island theorems established in related models. We identify two possible localization scenarii, depending on whether the exponent ruling the tail of the gap distribution is smaller or larger than one. This is joint work with François Simenhaus.