Abstract
We now have a good understanding how emergence occurs in the Fleming–Viot model with two fitness levels with one type at each level and fitness difference 1 between the two types (Section 8). It remains to determine in this Section 9 and the next Section 10 respectively what is the effect of having
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M ≥ 2 types on the lower level,
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M ≥ 2 types on the upper level.
Both of these cases are going to exhibit some new phenomenon and challenges not present in the two-type model. What are the effects of having more than two types precisely?
Keywords
- Mesoster
- Meso State
- Time Decoupling
- McKean Vlasov
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References
D. Aldous, J. Pitman, Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. Henri Poincare 34, 637–686 (1998)
K.B. Athreya, P. Ney, A renewal approach to Perron-Frobenius theory of non-negative kernels on general state spaces. Math. Zeit. 179, 507–529 (1982)
N.T.J. Bailey, The Elements of Stochastic Processes (Wiley, New York, 1964)
J. Bertoin, Random Fragmentation and Coagulation Processes (Cambridge University Press, Cambridge, 2006)
N.H. Bingham, R.A. Doney, Asmptotic properties of super-critical branching processes II: Crump-mode and Jirina processes. Adv. Appl. Probab. 7, 66–82 (1975)
P.J. Brockwell, J. Gani, S.I. Resnick, Birth, immigration and catastrophe processes. Adv. Appl. Probab. 14, 709–731 (1982)
P. Carmona, F. Petit, M. Yor, Some extensions of the arc sine law as (partial) consequences of the scaling property of Brownian motion. Probab. Theory Relat. Fields 100, 1–29 (1994)
J.B. Conway, A Course in Functional Analysis, 2nd edn. (Springer, Berlin, 1990)
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)
P. Diaconis, E. Mayer-Wolf, O. Zeitouni, M.P.W. Zerner, The Poisson-Dirichlet la is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32, 915–938 (2004)
R.A. Doney, A limit theorem for a class of supercritical branching processes. J. Appl. Probab. 9, 707–724 (1972)
R.A. Doney, On single and multi-type age-dependent branching processes. J. Appl. Probab. 13, 239–246 (1976)
S.N. Ethier, T.G. Kurtz, Markov Processes, Characterization and Convergence (Wiley, New York, 1986)
S.N. Ethier, T.G. Kurtz, Coupling and ergodic theorems for Fleming-Viot processes. Ann. Probab. 26, 533–561 (1998)
A. Göing-Jaeschke, M. Yor, A survey and some generalizations of Bessel process (1999), http://www.risklab.ch/papers.htm.
T.E. Harris, On the Theory of Branching Processes (Springer, Berlin, 1963)
P. Jagers, O. Nerman, The growth and composition of branching populations. Adv. Appl. Probab. 16, 221–259 (1984)
P. Jagers, O. Nerman, in The Asymptotic Composition of Supercritical, Multitype Branching Populations. Séminaire de probabilités (Strasbourg), tome 30 (1996), pp. 40–54
S. Karlin, H.M. Taylor, A First Course in Stochastic Processes, 2nd edn. (Academic, New York, 1975)
J.E. Marsden, On product formulas for nonlinear semigroups. J. Funct. Anal. 13, 51–74 (1973)
J. Martinez, J.M. Mazón, Quasi-compactness of dominated positive operators and C 0-semigroups. Math. Z. 207, 109–120 (1991)
S.-T.C. Moy, Extensions of a limit theorem of Everett, Ulm and Harris on multitype branching processes to a branching process with countably many types. Ann. Math. Stat. 38, 992–999 (1967)
S.-T.C. Moy, Ergodic properties of expectation matrices of a branching process with countably many types. J. Math. Mech. 16, 1207–1225 (1967)
O. Nerman, On the convergence of supercritical general (C-M-J) branching processes. Zeitschrift f. Wahrscheinlichkeitsth. verw. Gebiete 57, 365–395 (1981)
O. Nerman, On the Convergence of Supercritical General Branching Processes. Thesis, Department of Mathematics, Chalmers University of Technology and the University of Göteborg, 1979
P.S. Puri, Some further results on the birth-and-death process and its integral. Proc. Camb. Philos. Soc. 64, 141–154 (1968)
L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 2 (Wiley, New York, 1987)
H. Shirakawa, Squared Bessel processes and their applications to the square root interest rate model. Asia-Pac. Financ. Mark. 9, 169–190 (2002)
D. Vere-Jones, Ergodic properties of non-negative matrices, I, II. Pac. J. Math. 22, 361–386; 26, 601–620 (1967/1968)
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Dawson, D.A., Greven, A. (2014). Emergence with M ≥ 2 Lower Order Types (Phases 0,1,2). 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_9
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