Verzweigungsprozesse und Irrfahren in zufälliger Umgebung

• Applicant: Professor Dr. Götz Kersting
• Subject Area: Mathematics
• Term: from 2007 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 58528951

• Final Report Year: 2011

Final Report Abstract

Branching processes in random environment (BPRE) extend the classical Galton-Watson model for population growth in that the offspring distribution may vary randomly from one generation to the other. As a consequence the longtime behaviour of BPRE is much richer and exhibits phenomena not known from classical Galton-Watson processes. For a long time BPRE had been analysed mainly under the very special assumption of geometric offspring distributions. In this project (which continued our russian-german cooperation) it was our goal to handle general offspring distributions. This required to approach the processes in a more probabilistic way, Laplace transforms are no longer the only method of investigation. As it turned out, the assumptions on the associated random walk can also be considerably weakened. The main idea and hope of our project was that the different expertise of our Russian collegues on Laplace transforms and ours on tree constructions and other probabilistic approaches could be combined fruitfully. This hope has beautifully become true, our results would not have been obtained without this cooperation. A main purpose of our research was to obtain a proper understanding of the phase transition, which occurs for subcritical BPRE, conditioned on survival. Here we concentrated on weakly subcritical processes and on the intermediately subcritical case, which exhibit a rich behavior. We studied this phase transition also on the level of large deviations and within the context of diffusion approximations of the processes. Further results concern the application of BPRE to random walks in random environment.

Publications

• Sudden extinction of a critical branching process in a random environment. (Russian) Teor. Veroyatn. Primen. 54 417–438 (2009); translation in Theory Probab. Appl. 54 466–484 (2010)
  V. Vatutin, V. Wachtel
  
• A refinement of limit theorems for the critical branching processes in random environment. Workshop on Branching Processes and their Applications Lect. Notes Stat. Proc., 197, 3–19, Springer, Berlin (2010)
  V. Vatutin
  
• Branching processes in random environment which extinct at a given moment. Markov Process. rel. Fields 16, 329–350 (2010)
  C. Böinghoff, E. E. Dyakonova, G. Kersting, V. Vatutin
  
• Branching processes in random environment. Dissertation, (2010)
  C. Böinghoff
  
• Limit theorems for weakly subcritical branching processes in random environment. J. Theoretical Probab. (2010)
  V.I. Afanasyev, C. Böinghoff, G. Kersting, V. Vatutin
  
• On large deviations of branching processes in a random environment - Offspring distributions with geometrically bounded tails. Stoch. Process. Appl. 120, 2064–2077 (2010)
  C. Böinghoff and G. Kersting
  
• Branching diffusions in random environment. (2011)
  C. Böinghoff and M. Hutzenthaler
  
• Conditional limit theorems for intermediately subcritical branching processes in random environment. (2011)
  V.I. Afanasyev, C. Böinghoff, G. Kersting, V. Vatutin
  
• Upper large deviations for Branching Processes in Random Environment for heavy-tailed offspring distributions. Electr. J. Probab. 16, 1900–1933 (2011)
  V. Bansaye and C. Böinghoff