Irrfahrten auf freien Produkten, amalgamierten Produkten, Bäumen und Strings über unendlichem Alphabet

• Applicant: Privatdozent Dr. Lorenz A. Gilch
• Subject Area: Mathematics
• Term: from 2008 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 82859762

This project is devoted to the investigation of random walks on different structures, and to categorize structures in terms of the random walk. The structures studied are interrelated, and include free products, free products with amalgamation by an infinite subgroup, trees generated by infinite connected graphs, and strings over an infinite alphabet. The plan is to deepen the understanding of the behaviour of random walks on these structures by applying a wide range of mathematical techniques from probability theory (random walks), structure theory (algebra, geometry and graph theory) and analysis (potential theory). This includes elaborating criteria for transience and recurrence of random walks on these structures, investigating positivity of the rate of escape for transient random walks, computing explicit formulas for the rate of escape, deriving local limit theorems for transition probabilites, and investigating the boundary process. There are well-known results in the “finite” case, where the amalgamating subgroup is finite, trees have only a finite number of cone types, and strings are over a finite alphabet. The essence of this project is to study these questions in relation to the underlying structure and to introduce new techniques to attack these problems in the “infinite” case.

• DFG Programme: Research Fellowships
• International Connection: Austria
• Host: Professor Dr. Wolfgang Woess