Abstract
An important technical tool in our analysis are various new representations of the marginal distributions of the basic processes in terms of expectations under the appropriate function-valued respectively set-valued dual processes.
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References
S.R. Athreya, J.M. Swart, Branching-coalescing systems. Probab. Theory Relat. Fields 131(3), 376–414. Electronic, 39 p. (2005). doi:10.1007/s00440-004- 0377-4
D.A. Dawson, Measure-valued Markov processes, in École d’Été de Probabilités de Saint Flour XXI. Lecture Notes in Mathematics 1541 (Springer, Berlin, 1993), pp. 1–261
D.A. Dawson, Multilevel mutation-selection systems and set-valued duals (in preparation)
D.A. Dawson, A. Greven, Hierarchically interacting Fleming-Viot processes with selection and mutation: Multiple space time scale analysis and quasi equilibria. Electron. J. Probab. 4, paper no. 4, 1–81 (1999)
A. Depperschmidt, A. Greven, P. Pfaffelhuber, Tree-valued Fleming-Viot dynamics with mutation and selection. Ann. Appl. Probab. 22, 2560–2615 (2012)
A.M. Etheridge, in An Introduction to Superprocesses. (English summary). University Lecture Series, vol. 20 (American Mathematical Society, Providence, 2000)
A.M. Etheridge, R.C. Griffiths, A coalescent dual process in a Moran model with genic selection. Theor. Popul. Biol. 75, 320–330 (2009)
S.N. Ethier, T.G. Kurtz, Markov Processes, Characterization and Convergence (Wiley, New York, 1986)
P. Fernhead, Perfect simulation from population genetic models with selection. Theor. Popul. Biol. 59, 263–279 (2001)
A. Greven, V. Limic, A. Winter, Representation theorems for interacting Moran models, interacting Fisher–Wright diffusions and applications. Electron. J. Probab. 10(39), 1286–1358 (2005)
A. Greven, P. Pfaffelhuber, A. Winter, Tree-valued resampling dynamics: Martingale Problems and applications. Probab. Theor. Relat. Fields 155, 787–838 (2013)
A. Greven, P. Pfaffelhuber, A. Winter, Convergence in distribution of random metric measure spaces (The \(\Lambda\)-coalescent measure tree). Probab. Theor. Relat. Fields 145, 285–322 (2009)
P. Jagers, O. Nerman, The growth and composition of branching populations. Adv. Appl. Probab. 16, 221–259 (1984)
S. Krone, C. Neuhauser, Ancestral processes with selection. Theor. Popul. Biol. 51, 210–237 (1997)
T.M. Liggett, Interacting Particle Systems (Springer, New York, 1985)
T. Shiga, Diffusion processes in population genetics. J. Math. Kyoto Univ. 21-1, 133–151 (1981)
T. Shiga, Continuous time multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ. 22-1, 1–40 (1982)
T. Shiga, Uchiyama, Stationary states and the stability of the stepping stone model involving mutation and selection. Probab. Theory Relat. Fields 73, 87–117 (1986)
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Dawson, D.A., Greven, A. (2014). A Basic Tool: Dual Representations. In: Spatial Fleming-Viot Models with Selection and Mutation. Lecture Notes in Mathematics, vol 2092. Springer, Cham. https://doi.org/10.1007/978-3-319-02153-9_5
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DOI: https://doi.org/10.1007/978-3-319-02153-9_5
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