Annals of Probability

Self-interacting diffusions. III. Symmetric interactions

Michel Benaïm and Olivier Raimond

Full-text: Open access

Abstract

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to $$dX_{t}=dW_{t}(X_{t})-\frac{1}{t}\biggl(\int_{0}^{t}\nabla V_{X_{s}}(X_{t})\,ds\biggr)\,dt,$$ where {Wt} is a Brownian vector field on M and Vx(y)=V(x,y) a smooth function. Let $\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\,ds$ denote the normalized occupation measure of Xt. We prove that, when V is symmetric, μt converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μt converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.

Article information

Source
Ann. Probab., Volume 33, Number 5 (2005), 1716-1759.

Dates
First available in Project Euclid: 22 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1127395871

Digital Object Identifier
doi:10.1214/009117905000000251

Mathematical Reviews number (MathSciNet)
MR2165577

Zentralblatt MATH identifier
1085.60073

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 62L20: Stochastic approximation 37B25: Lyapunov functions and stability; attractors, repellers

Keywords
Self-interacting random processes reinforced processes

Citation

Benaïm, Michel; Raimond, Olivier. Self-interacting diffusions. III. Symmetric interactions. Ann. Probab. 33 (2005), no. 5, 1716--1759. doi:10.1214/009117905000000251. https://projecteuclid.org/euclid.aop/1127395871


Export citation

References

  • Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 337–404.
  • Benaïm, M. (1999). Dynamics of stochastic approximation algorithms. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 1–68. Springer, New York.
  • Benaïm, M., Ledoux, M. and Raimond, O. (2002). Self-interacting diffusions. Probab. Theory Related Fields 122 1–41.
  • Benaïm, M. and Raimond, O. (2002). On self-attracting/repelling diffusions. C. R. Acad. Sci. Sér. I 335 541–544.
  • Benaïm, M. and Raimond, O. (2003). Self-interacting diffusions II: Convergence in law. Ann. Inst. H. Poincaré 6 1043–1055.
  • 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.
  • Conley, C. C. (1978). Isolated Invariant Sets and the Morse Index. Amer. Math. Soc., Providence, RI.
  • Cucker, F. and Smale, S. (2001). On the mathematical foundations of learning. Bull. Amer. Math. Soc. 39 1–49.
  • Cranston, M. and Le Jan, Y. (1995). Self-attracting diffusions: Two cases studies. Math. Ann. 303 87–93.
  • Cranston, M. and Mountford, T. S. (1996). The strong law of large numbers for a Brownian polymer. Ann. Probab. 24 1300–1323.
  • Dieudonné, J. (1972). Eléments d'Analyse. I. Gauthier-Villars, Paris.
  • Durrett, R. T. and Rogers, L. C. G. (1992). Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 337–349.
  • Elworthy, K. D. and Tromba, A. J. (1970). Degree theory on Banach manifolds. In Nonlinear Functional Analysis 86–94. Amer. Math. Soc., Providence, RI.
  • Hermann, S. and Roynette, B. (2003). Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325 81–96.
  • Hirsch, M. W. and Pugh, C. C. (1970). Stable manifolds and hyperbolic sets. In Global Analysis 133–163. Amer. Math. Soc., Providence, RI.
  • Hofbauer, J. (2000). From Nash and Brown to Maynard Smith: Equilibria, dynamics, and ESS. Selection 1 81–88.
  • Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equation and Diffusion Processes. North-Holland, Amsterdam.
  • Irwin, M. C. (1970). On the stable manifold theorem. Bull. London Math. Soc. 2 196–198.
  • Lang, S. (1993). Real and Functionnal Analysis, 3rd ed. Springer, New York.
  • Malrieu, F. (2001). Inégalités de Sobolev logarithmiques pour des problèmes d'évolution non linéaires. Ph.D. thesis, Univ. Paul Sabatier, Toulouse III.
  • 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.
  • Pemantle, R. (2002). Random processes with reinforcement. Preprint.
  • Pemantle, R. (1990). Nonconvergence to unstable points in urn models and stochastic approximations. Ann. Probab. 18 698–712.
  • Raimond, O. (1996). Self-attracting diffusions: Case of the constant interaction. Probab. Theory Related Fields 107 177–196.
  • Schoenberg, I. J. (1938). Metric spaces and completely monotone functions. Ann. of Math. 39 811–841.
  • Smale, S. (1965). An infinite dimensional version of Sard's theorem. Amer. J. Math. 87 861–866.
  • Tarrès, P. (2000). Pièges répulsifs. C. R. Acad. Sci. Paris Sér. I 330 125–130.
  • Tarrès, P. (2001). Pièges des algorithmes répulsifs et marches aléatoires renforcées par sommets. Ph. D. dissertation, Ecole Normale.
  • Tromba, A. J. (1977). The Morse–Sard–Brown theorem for functionals and the problem of Plateau. Amer. J. Math. 99 1251–1256.