Asymptotic analysis and diffusion limit of the Persistent Turning Walker model

Asymptotic Analysis

Volumn 67, Issue 1-2, 2010, Pages 17-31

  Cattiaux, Patrick ^a   Chafaï, Djalil ^a,b   Motsch, Sébastien ^a  

^a UNIVERSITÉ DE TOULOUSE   (France)

^b UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

Animal behavior; Central limit theorems; Gaussian and Markov processes; Hypo elliptic diffusions; Invariance principles; Kinetic Fokker Planck equations; Mathematical Biology; Poisson equation

Indexed keywords

ANIMAL BEHAVIOR; CENTRAL LIMIT THEOREM; GAUSSIAN AND MARKOV PROCESSES; GAUSSIANS; HYPO-ELLIPTIC DIFFUSIONS; INVARIANCE PRINCIPLE; MATHEMATICAL BIOLOGY;

ANIMALS; ASYMPTOTIC ANALYSIS; DIFFUSION; EQUATIONS OF MOTION; GEOMETRY; MARKOV PROCESSES; POISSON EQUATION; RECONNAISSANCE AIRCRAFT; STOCHASTIC MODELS; TIMING JITTER; TURNING;

FOKKER PLANCK EQUATION;

EID: 77950255697     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: 10.3233/ASY-2009-0969     Document Type: Article

References (27)

1. C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les inégalité s de Sobolev logarithmiques, Panoramas et Synthèses, Vol.10, Société Mathématique de France, Paris, 2000. (With a preface by D. Bakry and M. Ledoux.)
2. D. Bakry, P. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Func. Anal. 254 (2008), 727-759.
3. E. Casellas, J. Gautrais, R. Fournier, S. Blanco, M. Combe, V. Fourcassié, G. Theraulaz and C. Jost, From individual to collective displacements in heterogeneous environments, Journal of Theoretical Biology 250(3) (2008), 424-434.
4. P. Cattiaux and A. Guillin, Deviation bounds for additive functionals of Markov processes, ESAIM Probability and Statistics 12 (2008), 12-29.
5. E.A. Codling, M.J. Plank and S. Benhamou, Random walk models in biology, Journal of The Royal Society Interface 5 (2008), 813-834.
6. P. Degond and S. Motsch, Large scale dynamics of the Persistent Turning Walker Model of fish behavior, Journal of Statistical Physics 131(6) (2008), 989-1021.
7. J.-D. Deuschel and D.W. Stroock, Large Deviations, Pure and Applied Mathematics, Vol.137, Academic Press, Boston, MA, 1989.
8. R. Douc, G. Fort and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Preprint, 2006. Available on Mathematics ArXiv.math.ST/0605791.
9. N. Gantert, J. Garnier, S. Olla, Zh. Shi and A.-S. Sznitman, Milieux aléatoires, Panoramas et Synthèses, F. Comets and É. Pardoux, eds, Vol.12, Société Mathématique de France, Paris, 2001.
10. J. Gautrais, C. Jost, M. Soria, A. Campo, S. Motsch, R. Fournier, S. Blanco and G. Theraulaz, Analyzing fish movement as a persistent turning walker, J. sMath. Biol. 58 (2008), 429-445.
11. M. Hairer and A.G. Pavliotis, Periodic homogenization for hypoelliptic diffusions, J. Stat. Phys. 117(1) (2004), 261-279.
12. I.S. Helland, Central limit theorem for martingales with discrete or continuous time, Scand. J. Statist. 9 (1982),79-94.
13. J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften, Vol.288, Springer-Verlag, Berlin, 2003.
14. R. Jeanson, S. Blanco, R. Fournier, J.L. Deneubourg, V. Fourcassié and G. Theraulaz, A model of animal movements in a bounded space, Journal of Theoretical Biology 225(4) (2003), 443-451
15. G.L. Jones, On the Markov chain central limit theorem, Probab. Surv. 1 (2004), 299-320 (electronic).
16. U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, Vol.6, Walter de Gruyter, Berlin, 1985. (With a supplement by Antoine Brunel.)
17. Y.A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer Series in Statistics, Springer-Verlag, London, 2004.
18. S. Méléard, Asymptotic behaviour of some interacting particle systems: McKean-Vlasov and Boltzmann models, in: Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Lecture Notes in Math., Vol.1627, Springer-Verlag, Berlin, 1996, pp. 42-95.
19. E. Pardoux and A.Y. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Prob. 29(3) (2001), 1061-1085.
20. E. Pardoux and A.Y. Veretennikov, On the Poisson equation and diffusion approximation. II, Ann. Prob. 31(3) (2003), 1116-1192.
21. E. Pardoux and A.Y. Veretennikov, On the Poisson equation and diffusion approximation. III, Ann. Prob. 33(3) (2005), 1111-1133.
22. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edn, Grundlehren der Mathematischen Wissenschaften, Vol.293, Springer-Verlag, Berlin, 1999.
23. G. Royer, Une initiation aux inégalités de Sobolev logarithmiques, Société Mathématique de France, Paris, 1999.
24. A.-S. Sznitman, Topics in propagation of chaos, in: École d'Été de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math., Vol.1464, Springer-Verlag, Berlin, 1991, pp. 165-251.
25. G. Theraulaz, E. Bonabeau, S.C. Nicolis, R.V. Sole, V. Fourcassié, S. Blanco, R. Fournier, J.L. Joly, P. Fernandez, A. Grimal et al., Spatial patterns in ant colonies, Proceedings of the National Academy of Sciences 99(15) (2002), 9645.
26. T. Vicsek, A question of scale, Nature 411(6836) (2001), 421.
27. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147(3) (1998), 585-650.