# Complex Stochastic Systems: Discrete vs. Continuous

Hausdorff Trimester Program

September 2007 - February 2008

**Organizers:** Sergio Albeverio, Jennifer Chayes, Gerard Ben Arous, Michel Ledoux, Karl Theodor Sturm

Applied stochastic models are frequently very complex, high or infinite dimensional, with a mixture of discrete and continuous elements, singularities and nonlinearities. Their mathematical description and analysis often requires a combination of methods from discrete probability and combinatorics, stochastic analysis, partial differential equations, and geometric analysis. For example, discretization of the continuous components of a stochastic model is fundamental for numerical simulations. Conversely, continuum limits can be helpful to gain a better understanding of discrete models, e.g. in statistical mechanics. Powerful techniques, e.g. logarithmic Sobolev inequalities and concentration of measure estimates have originally been developed in a continuous setup, partially motivated by geometric considerations. Nowadays they are of rapidly increasing importance also in discrete setups, in the analysis of asymptotic and non-asymptotic issues of stochastic processes like the rate of convergence to equilibrium, and for various algorithmic applications. The extension of concepts from geometric stochastic analysis to singular spaces, graphs and random structures is a major challenge. A better understanding of the fascinating connections between random matrices and number theory requires a combination of probability, geometric analysis and algebra.

Hence there are clear needs to increase significantly the interaction between stochastic analysis and non-linear partial differential equations on the one side, and discrete probability and theoretical computer science on the other side. The aim of this program was to bring together scientists from these different communities, and thus to boost new developments. A particular emphasis was given to training of postdocs and PhD students in this important field. This is reflected in a high number of introductory courses by leading top-level scientists. The program also addressed to young researchers from neighboring fields such as analysis and computer science.