### Workshop in honor of Herbert Spohn and 11th Probability Day Erlangen-München:

The workshop will take place on Friday, June 29 und on Saturday, June 30, 2012. There will be 3 talks on Friday afternoon and 3 talks on Saturday morning.

The talks on Friday will take place at the Technical University of Munich, Zentrum Mathematik, Boltzmannstr. 3, 85748 Garching, in room MI 00.07.011. The room is located on the ground floor. How to go to there?

The talks on Saturday will take place at the University of Munich, Mathematisches Institut, Theresienstrasse 39, 80333 München, in room A 027. How to go to there?

### Program:

• Friday, June 29
• 13:45-14:00 Opening
• 15:00-15:45 Giambattista Giacomin: Active rotator models: a bridge between nonequilibrium statistical mechanics and collective phenomena in biology
• 16:30-17:15 Errico Presutti: Fourier law and moving interfaces in the SEP
• Saturday, June 30
• 9:30-10:30 Patrik Ferrari: t.b.a.
• 10:30-11:15 Frank den Hollander: Renormalisation of hierarchically interacting Cannings processes
• 12:00-12:45 Jean-Dominique Deuschel: t.b.a.

### Titles and abstract

• Giambattista Giacomin: Active rotator models: a bridge between nonequilibrium statistical mechanics and collective phenomena in biology
Abstract: Active rotator models have been introduced in the life sciences, more precisely in the neurosciences, as minimal models of interacting 'excitable' systems exhibiting time periodic behaviors in the limit of infinitely many interacting units. They are mean field type models of interacting diffusions on the unit circle and, in the most interesting and biologically relevant cases, they are non-reversible. The oscillatory behavior sets in as the result of the combined effect of noise and interaction and it is intimately related to the non-reversible character of model. The approach we present is based on the observation that active rotator models reduce, for a particular choice of the parameters, to a reversible Langevin dynamics for the mean field plane rotator (or classical XY spin) model. The analysis is carried out at the level of the Fokker-Planck PDE for the evolution of the system's empirical density, in the limit of infinitely many interacting units. Such a Fokker-Planck PDE is the gradient flow of a free energy when the underlying dynamics is reversible and we will exploit the sharp control we have on this PDE to obtain results on the nonreversible/non-gradient flow case that interests us.
• Frank den Hollander: Renormalisation of hierarchically interacting Cannings processes
Abstract: In order to analyse universal patterns in the large space-time behaviour of interacting multi-type genetic populations, a key approach has been to carry out a renormalisation analysis. This has provided considerable insight into the genealogical structure of such populations.
In this talk we describe a system of hierarchically interacting Cannings processes. The latter are jump process that arise as the continuum limit of a population model in which the offspring of a single individual can be a positive fraction of the total population. The interaction between the individuals comes from migration and resampling on all hierarchical space-time scales simultaneously. Individuals live in colonies labelled by the hierarchical group $Omega_N$ of order N, and are subject to migration and resampling in k-blocks for all $k\in\N_0$ based on a sequence of migration rates $\uc=(c_k)_{k\in\N_0}$ and a sequence of offspring measures $\uL= (\Lambda_k)_{k\in\N_0}$. For this system we carry out a full renormalisation analysis in the hierarchical mean-field limit $N \to \infty$. Our main result is that, in the limit as $N\to\infty$, on each hierarchical scale $k\in\N_0$ the k-block averages of the system converge to a random process that is a superposition of a single-component Cannings process and a single-component Fleming-Viot diffusion, the latter with a drift $c_k$ and with a volatility constant $d_k$ that turns out to be a function of $c_l$ and $\Lambda_l$ for all $0 \leq l <k$. It is through the volatility that the renormalisation manifests itself.
We investigate how the volatility scales as $k\to\infty$, which requires an analysis of iterations of certain Möbius-transformations. We discuss the implications of this scaling for the behaviour on large space-time scales, comparing the outcome with what is known from the renormalisation analysis of hierarchically interacting Fleming-Viot diffusions, pointing out several new features.
(joint work with Andreas Greven, Sandra Kliem and Anton Klimovsky)
• Errico Presutti: Fourier law and moving interfaces in the SEP
Abstract: I shall present some recent results (in collaboration with A. De Masi, D. Tsagkarogiannis and M.E. Vares) and conjectures (with some preliminary results in collaboration also with K. Ravishankar) on the one dimensional symmetric exclusion process (SEP) with birth-death processes at the fixed (or moving) boundaries which simulate "current reservoirs".

How to get to Garching-Hochbrück