Tree-valued stochastic processes

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Classically, allelic frequencies in large populations are modelled as (real- or measure-valued) diffusion processes (see e.g. Etheridge, A. An introduction to superprocesses. American Mathematical Society, 2001). Much of their analysis is done using a backward, i.e., coalescent-based approach. We integrate these two approaches into a single stochastic process, taking values in genealogical trees. By now, we have constructed tree-valued processes under neutrality (Greven et al, 2010) and under selection (Depperschmidt et al. 2011). Topological foundations are laid in Greven et al. (2009) and Depperschmidt et al. (2011). In the selective case, our analysis gives for the first time approximations for the genealogical distances in a sample in equilibrium. In future work, we address the biologically important consequences of recombination on the evolution of genealogies.

Additional information: http://www.stochastik.uni-freiburg.de
Phone: 0761/203-5667
Email: peter.pfaffelhuber@stochastik.uni-freiburg.de

Start of project: 2005
End of project: (unlimited)

Albert-Ludwigs-University Freiburg
Prof. Dr. Peter Pfaffelhuber

Actual Research Report

• Depperschmidt A

• Greven A, Pfaffelhuber P, Winter A: Tree-valued resampling dynamics Martingale problems and applications Probab Theory Rel, 2013; 155: 789-838. : https://doi.org/10.1007/s00440-012-0413-8 (download: http://arxiv.org/abs/0806.2224)
• Greven A, Pfaffelhuber P, Winter A: Convergence in distribution of random metric measure spaces: (Lambda-coalescent measure trees). Probab Theory Rel, 2009; 145 (1): 285-322. : http://dx.doi.org/10.1007/s00440-008-0169-3 (download: http://arxiv.org/abs/math/0609801)