ORGANIZER

SPEAKERS

PROGRAMME

WEDNESDAY OCTOBER 10

ABSTRACTS

Roberto Fernandez

A regular process that is non gibbssian

Processes are determined by transition probabilities, that is by conditional expectations with respect to the past. In contrast, one-dimensional Gibbs measures are fields determined by simultaneous conditioning on past and future. For Markovian and exponentially continuous processes both theories are known to be equivalent.
We present a simple process showing that this equivalence does not extend to more general cases. The process is ergodic, has a continuous dependence with respect to the past and even admits a renewal construction. Yet, a straightforward explicit calculation shows that it is not Gibbsian.
(joint work with G. Maillard (Marseille) and S. Gallo (U. Federal do Rio de Janeiro, Brazil)

Frank den Hollander

Renormalization of hierarchically interacting Cannings processes

In order to analyse universal patterns in the large space-time behaviour of interacting multi-type stochastic populations, a key approach has been to carry out a renormalization analysis in the hierarchical mean-field limit. This has provided considerable insight into the structure of interacting systems of diffusions.
In this talk we bring a new class of interacting jump processes into focus, namely, hierachically interacting Cannings processes. Here, individuals in a population are organized into colonies, labelled by the hierarchical group of order $N$, and are subject to migration, reshuffling and resampling on all hierarchical scales simultaneously, at rates that depend on the scale.
For this system we carry out a full renormalisation analysis in the hierarchical mean-field limit $N \to \infty$. Our main results include a new classification for when the process exhibits clustering (= develops spatially expanding mono-type regions), respectively, exhibits local coexistence (= allows for different types to live next to each other with positive probability). In the clustering regime we find a rich scenario for the speed at which mono-type clusters grow in time.
(joint work with Andreas Greven, Sandra Kliem, Anton Klimovsky)

Errico Presutti

Symmetric simple exclusion process (SSEP) with births and deaths and convergence to a Stefan problem

I will describe some preliminary results obtained in collaboration with  Anna De Masi and Pablo Ferrari on the symmetric simple exclusion process (SSEP) on Z with births and deaths.  We consider configurations where there are finitely many particles to the right of the origin and finitely many holes to its left.  The evolution

 is SSEP at rate 1/2 plus   birth death processes each one at rate $\epsilon$,. The birth is given by the first hole being replaced by a particle and the death by the last particle being replaced by  a hole.  We prove the existence of a unique invariant measure as seen from the first hole and prove that the average distance between last particle and first hole scale as $\epsilon^{-1}$.  We also show that the macroscopic limit is described by a Stefan (free boundary) problem.

Frank Redig

Dynamics of the condensate in the symmetric inclusion process

The symmetric inclusion process is a particle system where particles perform random walks on a lattice, and interact by attracting each other. If, on a fixed finite lattice, the rate of random walk is converging to zero, and at the same time many particles are in the system, condensation phenomena occur. We study the asymptotic dynamics of these condensates. In the fully connected case, a single condensate is formed immediately, and hops randomly over the lattice. In the non fully connected case, several condensates form and merge according to a diffusive dynmics to form eventually a single condensate. 

PRACTICAL INFORMATION

CONTACT
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Last updated 03-10-12,
by PK