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) no. 6, pp. 1038-1077.

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.

Classification : 60J60, 60J70, 60K35
Mots clés : interacting diffusions, space-time renormalization, two-type populations, independent branching, catalytic branching, mutually catalytic branching, universality

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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