On the classification of pointed fusion categories up to weak Morita equivalence

A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories $C$ and $D$ are weakly Morita equivalent if there exists an indecomposable right module category $M$ over $C$ such that $Fun_C(M,M)$ and $D$ are tensor equivalent. We use the Lyndon-Hochschild-Serre spectral sequence associated to abelian group extensions to give necessary and sufficient conditions in terms of cohomology classes for two pointed fusion categories to be weakly Morita equivalent. This result may permit to classify the equivalence classes of pointed fusion categories of any given global dimension.Comment: Corrected version. Accepted for publication in PJM. 26 page

Bridges of L\'{e}vy processes conditioned to stay positive

We consider Kallenberg's hypothesis on the characteristic function of a L\'{e}vy process and show that it allows the construction of weakly continuous bridges of the L\'{e}vy process conditioned to stay positive. We therefore provide a notion of normalized excursions L\'{e}vy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the L\'{e}vy process conditioned to stay positive by using the weakly continuous bridges of the L\'{e}vy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned L\'{e}vy process, the Durrett-Iglehart theorem stating that the Brownian bridge from $0$ to $0$ conditioned to remain above $-\varepsilon$ converges weakly to the Brownian excursion as $\varepsilon \to0$, is extended to L\'{e}vy processes. We also extend the Denisov decomposition of Brownian motion to L\'{e}vy processes and their bridges, as well as Vervaat's classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ481 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation

We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem.Comment: 23 pages, 4 figures, AMSLaTeX (PDFLaTeX), accepted in Annales de l'Institut Henri Poincar\'e (B