Self-interacting diffusions. III. Symmetric interactions

Annals of Probability

Volumn 33, Issue 5, 2005, Pages 1716-1759

  Benaïm, Michel ^a   Raimond, Olivier ^a,b  

^a UNIVERSITY OF NEUCHÂTEL   (Switzerland)

^b UNIVERSITÉ PARIS SUD   (France)

Author keywords

Reinforced processes; Self interacting random processes

Indexed keywords

EID: 27644592800     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117905000000251     Document Type: Article

References (29)

1. ARONSZAJN, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 337-404.
2. BENAÏM, M. (1999). Dynamics of stochastic approximation algorithms. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 1-68. Springer, New York.
3. BENAÏM, M., LEDOUX, M. and RAIMOND, O. (2002). Self-interacting diffusions. Probab. Theory Related Fields 122 1-41.
4. BENAÏM, M. and RAIMOND, O. (2002). On self-attracting/repelling diffusions. C. R. Acad. Sci. Sér. I 335 541-544.
5. BENAÏM, M. and RAIMOND, O. (2003). Self-interacting diffusions II: Convergence in law. Ann. Inst. H. Poincaré 6 1043-1055.
6. CARRILLO, J. A., MCCANN, R. J. and VILLANI, C. (2003). Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Preprint.
7. CONLEY, C. C. (1978). Isolated Invariant Sets and the Morse Index. Amer. Math. Soc., Providence, RI.
8. CUCKER, F. and SMALE, S. (2001). On the mathematical foundations of learning. Bull. Amer. Math. Soc. 39 1-49.
9. CRANSTON, M. and LE JAN, Y. (1995). Self-attracting diffusions: Two cases studies. Math. Ann. 303 87-93.
10. CRANSTON, M. and MOUNTFORD, T. S. (1996). The strong law of large numbers for a Brownian polymer. Ann. Probab. 24 1300-1323.
11. DIEUDONNÉ, J. (1972). Eléments d'Analyse. I. Gauthier-Villars, Paris.
12. DURRETT, R. T. and ROGERS, L. C. G. (1992). Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 337-349.
13. ELWORTHY, K. D. and TROMBA, A. J. (1970). Degree theory on Banach manifolds. In Nonlinear Functional Analysis 86-94. Amer. Math. Soc., Providence, RI.
14. HERMANN, S. and ROYNETTE, B. (2003). Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325 81-96.
15. HIRSCH, M. W. and PUGH, C. C. (1970). Stable manifolds and hyperbolic sets. In Global Analysis 133-163. Amer. Math. Soc., Providence, RI.
16. HOFBAUER, J. (2000). From Nash and Brown to Maynard Smith: Equilibria, dynamics, and ESS. Selection 1 81-88.
17. IKEDA, N. and WATANABE, S. (1981). Stochastic Differential Equation and Diffusion Processes. North-Holland, Amsterdam.
18. IRWIN, M. C. (1970). On the stable manifold theorem. Bull. London Math. Soc. 2 196-198.
19. LANG, S. (1993). Real and Functionnal Analysis, 3rd ed. Springer, New York.
20. MALRIEU, F. (2001). Inégalités de Sobolev logarithmiques pour des problèmes d'évolution non linéaires. Ph.D. thesis, Univ. Paul Sabatier, Toulouse III.
21. NORRIS, J. R., ROGERS, L. C. G. and WILLIAMS, D. (1987). Self-avoiding random walk: A Brownian motion model with local time drift. Probab. Theory Related Fields 74 271-287.
22. PEMANTLE, R. (2002). Random processes with reinforcement. Preprint.
23. PEMANTLE, R. (1990). Nonconvergence to unstable points in urn models and stochastic approximations. Ann. Probab. 18 698-712.
24. RAIMOND, O. (1996). Self-attracting diffusions: Case of the constant interaction. Probab. Theory Related Fields 107 177-196.
25. SCHOENBERG, I. J. (1938). Metric spaces and completely monotone functions. Ann. of Math. 39 811-841.
26. SMALE, S. (1965). An infinite dimensional version of Sard's theorem. Amer. J. Math. 87 861-866.
27. TARRÈS, P. (2000). Pièges répulsifs. C. R. Acad. Sci. Paris Sér. I 330 125-130.
28. TARRÈS, P. (2001). Pièges des algorithmes répulsifs et marches aléatoires renforcées par sommets. Ph. D. dissertation, Ecole Normale.
29. TROMBA, A. J. (1977). The Morse-Sard-Brown theorem for functionals and the problem of Plateau. Amer. J. Math. 99 1251-1256.