The renormalization transformation for two-type branching models
Dawson, D. A. ; Greven, A. ; den Hollander, F. ; Sun, Rongfeng ; Swart, J. M.
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 1038-1077 / Harvested from Numdam
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Cet article étudie des systèmes dénombrables de diffusions en interaction hiérarchiques et linéaires vivant dans le quadrant positif. De tels systèmes apparaissent dans la dynamique d'individus de deux types qui migrent tout en interagissant dans des colonies. Le comportement à grande échelle et temps long peut être étudié en utilisant le programme de renormalisation. Ce programme, qui a permis de résoudre d'autres cas (principalement uni-dimensionnels) est basé sur la construction et l'analyse d'une transformation de renormalisation non linéaire, agissant sur la fonction de diffusion des composants du système et connectant l'évolution de blocs moyennés sur le temps à différentes échelles. Nous identifions une classe générale de fonctions de diffusion dans le quadrant positif pour lequel la transformation de renormalisation est bien définie et qui, sous une conjecture de comportement aux bords, peut-être itérée. À l'intérieur de certaines sous-classes, nous identifiens les points fixes de la transformation et étudions leurs domaines d'attraction. Ces domaines d'attraction constitutent les classes d'universalité du système après changement d'échelle dans le temps et l'espace.

This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space-time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space-time scaling.

Publié le : 2008-01-01
DOI : https://doi.org/10.1214/07-AIHP143
Classification:  60J60,  60J70,  60K35 
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     author = {Dawson, D. A. and Greven, A. and den Hollander, Frank and Sun, Rongfeng and Swart, J. M.},
     title = {The renormalization transformation for two-type branching models},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
     pages = {1038-1077},
     doi = {10.1214/07-AIHP143},
     mrnumber = {2469334},
     zbl = {1181.60122},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_6_1038_0}
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Dawson, D. A.; Greven, A.; den Hollander, F.; Sun, Rongfeng; Swart, J. M. The renormalization transformation for two-type branching models. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 1038-1077. doi : 10.1214/07-AIHP143. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_6_1038_0/

[1] S. R. Athreya, M. T. Barlow, R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Related Fields 123 (2002) 484-520. | MR 1921011 | Zbl 1007.60053

[2] J.-B. Baillon, P. Clément, A. Greven and F. Den Hollander. On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions, Part I: The compact case. Canad. J. Math. 47 (1995) 3-27. | MR 1319687 | Zbl 0832.60097

[3] J.-B. Baillon, P. Clément, A. Greven and F. Den Hollander. On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions, Part II: The non-compact case. J. Funct. Anal. 146 (1997) 236-298. | MR 1446381 | Zbl 0873.60073

[4] R. F. Bass. Diffusions and Elliptic Operators. Springer, New York, 1998. | MR 1483890 | Zbl 0914.60009

[5] R. F. Bass and E. A. Perkins. Countable systems of degenerate stochastic differential equations with applications to super-Markov chains. Electron. J. Probab. 9 (2004) 634-673. | MR 2110015 | Zbl 1067.60037

[6] R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations arising from catalytic branching networks. Preprint. | MR 2448130

[7] J. T. Cox, D. A. Dawson and A. Greven. Mutually Catalytic Super Branching Random Walks: Large Finite Systems and Renormalization Analysis. Amer. Math. Soc., Providence, RI, 2004. | MR 2074427 | Zbl 1063.60143

[8] J. T. Cox and A. Greven. Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab. 22 (1994) 833-853. | MR 1288134 | Zbl 0806.60100

[9] D. A. Dawson, L. G. Gorostiza and A. Wakolbinger. Degrees of transience and recurrence and hierarchical random walks. Potential Anal. 22 (2005) 305-350. | MR 2135263 | Zbl 1075.60043

[10] D. A. Dawson and A. Greven. Multiple scale analysis of interacting diffusions. Probab. Theory Related Fields 95 (1993) 467-508. | MR 1217447 | Zbl 0791.60094

[11] D. A. Dawson and A. Greven. Hierarchical models of interacting diffusions: Multiple time scales, phase transitions and cluster formation. Probab. Theory Related Fields 96 (1993) 435-473. | MR 1234619 | Zbl 0794.60101

[12] D. A. Dawson and A. Greven. Multiple space-time analysis for interacting branching models. Electron. J. Probab. 1 (1996) 1-84. | MR 1423467 | Zbl 0890.60093

[13] D. A. Dawson, A. Greven and J. Vaillancourt. Equilibria and quasi-equilibria for infinite collections of interacting Fleming-Viot processes. Trans. Amer. Math. Soc. 347 (1995) 2277-2360. | MR 1297523 | Zbl 0831.60102

[14] D. A. Dawson, A. Greven and I. Zähle. Continuum limits of multitype population models on the hierarchical group. In preparation.

[15] D. A. Dawson and P. March. Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems. J. Funct. Anal. 132 (1995) 417-472. | MR 1347357 | Zbl 0853.60043

[16] D. A. Dawson and E. A. Perkins. Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26 (1998) 1088-1138. | MR 1634416 | Zbl 0938.60042

[17] D. A. Dawson and E. A. Perkins. On the uniqueness problem for catalytic branching networks and other singular diffusions. Illinois J. Math. 50 (2006) 323-383. | MR 2247832 | Zbl 1107.60045

[18] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. | MR 1609153 | Zbl 0709.60002

[19] S. N. Ethier and T. G. Kurtz. Markov Processes - Characterization and Convergence. Wiley, New York, 1986. | MR 838085 | Zbl 0592.60049

[20] K. Fleischmann and A. Greven. Diffusive clustering in an infinite system of hierarchically interacting Fisher-Wright diffusions. Probab. Theory Related Fields 98 (1994) 517-566. | MR 1271108 | Zbl 0794.60107

[21] K. Fleischmann and J. M. Swart. Renormalization analysis of catalytic Wright-Fisher diffusions. Electron. J. Probab. 11 (2006) 585-654. | MR 2242657 | Zbl 1113.60082

[22] A. Greven. Renormalization and universality for multitype population models. In Interacting Stochastic Systems 209-246. J.-D. Deuschel and A. Greven, Eds. Springer, Berlin, 2005. | MR 2118576 | Zbl 1078.92065

[23] F. Den Hollander. Renormalization of interacting diffusions. In Complex Stochastic Systems 219-233. O. E. Barndorff-Nielsen, D. R. Cox and C. Klüppelberg, Eds. Chapman & Hall, Boca Raton, 2001. | MR 1893414 | Zbl 0983.60094

[24] F. Den Hollander and J. M. Swart. Renormalization of hierarchically interacting isotropic diffusions. J. Stat. Phys. 93 (1998) 243-291. | MR 1656371 | Zbl 0946.60094

[25] S. Kliem. Degenerate stochastic differential equations for catalytic branching networks. Preprint. Available at arXiv:0802.0035v1.

[26] R. G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge Univ. Press, 1995. | MR 1326606 | Zbl 0858.31001

[27] S. Sawyer and J. Felsenstein. Isolation by distance in a hierarchically clustered population. J. Appl. Probab. 20 (1983) 1-10. | MR 688075 | Zbl 0514.92013

[28] T. Shiga and A. Shimizu. Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 (1980) 395-416. | MR 591802 | Zbl 0462.60061

[29] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer, New York, 1979. | MR 532498 | Zbl 0426.60069

[30] J. M. Swart. Clustering of linearly interacting diffusions and universality of their long-time distribution. Probab. Theory Related Fields 118 (2000) 574-594. | MR 1808376 | Zbl 0981.60094

[31] J. M. Swart. Uniqueness for isotropic diffusions with a linear drift. Probab. Theory Related Fields 128 (2004) 517-524. | MR 2045951 | Zbl 1042.60050

[32] S. R. S. Varadhan. Probability Theory. Amer. Math. Soc., Providence, RI, 2001. | Zbl 0980.60002