10. Erlanger Münchner Tag der Stochastik

Veranstalter: Nina Gantert (München), Hans-Otto Georgii (München), Andreas Greven (Erlangen), Gerhard Keller (Erlangen), Franz Merkl (München), Silke Rolles (München), Vitali Wachtel (München).

Der Tag der Stochastik findet statt am Freitag, den 01.07.2011, im Raum 349 im Mathematischen Institut der Ludwig-Maximilian-Universität München. Information zu Anfahrt erhalten Sie hier. Vom Hauptbahnhof ist das Institut in ca. 25 Minuten zu Fuß zu erreichen.

Programm:

  • 14:00 - 15:00 Uhr: Rob van den Berg (Amsterdam): Extensions of the BK inequality.
    Abstract: The BK inequality, proved by van den Berg and Kesten in 1984 says that, for product measures on {0,1}^n, the probability that two increasing events 'occur disjointly' is smaller than or equal to the product of the two individual probabilities. This result is often used in percolation and interacting particle systems. Their conjecture that the inequality even holds for all events was proved by Reimer in 1994.
    In spite of Reimer's work, several natural, fundamental problems in this area remained open. During this talk I will discuss some very recent progress, in particular an extension of the BK inequality to randomly drawn subsets of fixed size (joint work with Johan Jonasson). I will also mention a modified version of the notion 'disjoint occurrence' for the Ising model (work in progress with Alberto Gandolfi).
  • 15:00 - 16:00 Uhr: Jiří Černý (Zürich): Vacant set of random walk on (random) graphs. 
    Abstract: The vacant set is the set of vertices not visited by a random walk on a graph before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the 'thermodynamic limit' when |G| and T grow simultaneously), and the relation of the problem to the random interlacement percolation. I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.
  • 16:00 - 16:30 Uhr: Kaffee und Tee.
  • 16:30 - 17:30 Uhr: Frank Aurzada (Berlin): The one-sided exit problem for integrated processes and fractional Brownian motion.
    Abstract: We study the one-sided exit problem problem, also known as one-sided barrier problem, that is, given a stochastic process X we would like to find, as T→∞, the asymptotic rate of P(sup_{0≤t≤T} X_t ≤ 1). This question is considered for alpha-fractionally integrated centered Lévy processes and 'integrated' centered random walks. We show that the rate of decrease of the above probability is polynomial with exponent θ=θ(α)>0 which only depends on alpha but not on the choice of the Lévy process or random walk.
    This generalizes results of Y.G. Sinai (1991) who considered α=1 and the simple random walk. Similar recent results are due to V. Vysotsky (2010) and A. Dembo and F. Gao (2011).
    Finally, the results are compared to the corresponding ones for fractional Brownian motion.
  • Nach den Vorträgen wird es ein Abendessen geben.