The aim of this paper is to study the long-term behavior of a class of self-interacting diffusion processes on ℝd. These are solutions to SDEs with a drift term depending on the actual position of the process and its normalized occupation measure μt. These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods. We extend these methods to ℝd, assuming a confinement potential satisfying some conditions. These hypotheses on the confinement potential are required since in general the process can be transient, and is thus very difficult to analyze. Finally, we illustrate our study with an example on ℝ2.
Le but de cet article est d'étudier le comportement asymptotique d'une classe de processus en auto-interaction sur ℝd. Ces processus de diffusion s'écrivent comme solution d'E.D.S. dont le terme de dérive dépend à la fois de la position actuelle du processus et de sa mesure empirique μt. Jusqu'à présent, Benaïm, Ledoux et Raimond ont mené l'étude de ce type de diffusions sur des espaces compacts, via des méthodes d'approximation stochastique. Nous étendons ces techniques à ℝd, en supposant l'existence d'un potentiel de confinement (vérifiant certaines conditions). Nous avons besoin de ces hypothèses sur le potentiel de confinement, car, en général, un tel processus peut être transient. Nous concluons cet article par un exemple sur ℝ2.
Mots-clés : self-interaction diffusion, reinforced processes, stochastic approximation
@article{AIHPB_2010__46_3_618_0, author = {Kurtzmann, Aline}, title = {The {ODE} method for some self-interacting diffusions on $\mathbb {R}^d$}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {618--643}, publisher = {Gauthier-Villars}, volume = {46}, number = {3}, year = {2010}, doi = {10.1214/09-AIHP206}, zbl = {1215.60056}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/09-AIHP206/} }
TY - JOUR AU - Kurtzmann, Aline TI - The ODE method for some self-interacting diffusions on $\mathbb {R}^d$ JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2010 SP - 618 EP - 643 VL - 46 IS - 3 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/09-AIHP206/ DO - 10.1214/09-AIHP206 LA - en ID - AIHPB_2010__46_3_618_0 ER -
%0 Journal Article %A Kurtzmann, Aline %T The ODE method for some self-interacting diffusions on $\mathbb {R}^d$ %J Annales de l'I.H.P. Probabilités et statistiques %D 2010 %P 618-643 %V 46 %N 3 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/09-AIHP206/ %R 10.1214/09-AIHP206 %G en %F AIHPB_2010__46_3_618_0
Kurtzmann, Aline. The ODE method for some self-interacting diffusions on $\mathbb {R}^d$. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, pp. 618-643. doi : 10.1214/09-AIHP206. http://archive.numdam.org/articles/10.1214/09-AIHP206/
[1] L'hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability Theory and Statistics. Ecole de Prob. de St-Flour 1-114. Springer, Berlin, 1994. | MR | Zbl
.[2] Diffusions hypercontractives. In Séminaire de Probabilités XIX 177-206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | EuDML | Numdam | MR | Zbl
and .[3] Dynamics of stochastic approximation algorithms. In Séminaire de Probabilités XXXIII 1-68. Lecture Notes in Math. 1709. Springer, Berlin, 1999. | EuDML | Numdam | MR | Zbl
.[4] Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dynam. Differential Equation 8 (1996) 141-176. | MR | Zbl
and .[5] Self-interacting diffusions. Probab. Theory Related Fields 122 (2002) 1-41. | MR | Zbl
, and .[6] Self-interacting diffusions III: Symmetric interactions. Ann. Probab. 33 (2005) 1716-1759. | MR | Zbl
and .[7] Self-attracting diffusions: Two cases studies. Math. Ann. 303 (1995) 87-93. | EuDML | MR | Zbl
and .[8] The strong law of large numbers for a Brownian polymer. Ann. Probab. 2 (1996) 1300-1323. | MR | Zbl
and .[9] Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59 (1984) 335-395. | MR | Zbl
and .[10] Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 (1992) 337-349. | MR | Zbl
and .[11] Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325 (2003) 81-96. | MR | Zbl
and .[12] Spectral theory and limit theory for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003) 304-362. | MR | Zbl
and .[13] Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge, 1990. | MR | Zbl
.[14] The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) IX (2000) 305-366. | Numdam | MR | Zbl
.[15] A convexity principle for interacting gases. Adv. Math. 128 (1997) 153-179. | MR | Zbl
.[16] Markov Chains and Stochastic Stability. Springer, London, 1993. | MR | Zbl
and .[17] An asymptotic result for Brownian polymers. Ann. Inst. H. Poincaré, Probab. Statist. 44 (2008) 29-46. | Numdam | MR | Zbl
and .[18] A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1-79. | MR | Zbl
.[19] Self-attracting diffusions: Case of the constant interaction. Probab. Theory Related Fields 107 (1997) 177-196. | MR | Zbl
.[20] Supercontractivity and ultracontractivity for (nonsymmetric) diffusions semigroups on manifolds. Forum Math. 15 (2003) 893-921. | MR | Zbl
and .[21] Pièges répulsifs. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 125-130. | Zbl
.[22] The Morse-Sard-Brown theorem for functionals and the problem of Plateau. Amer. J. Math. 99 (1977) 1251-1256. | MR | Zbl
.[23] Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl
.[24] Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 (1997) 417-424. | MR | Zbl
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