Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit

Samuel Herrmann (École des Mines Nancy)
Julian Tugaut (Université de Nancy)

Abstract


Self-stabilizing diffusions are stochastic processes, solutions of nonlinear stochastic differential equation, which are attracted by their own law. This specific self-interaction leads to singular phenomenons like non uniqueness of associated stationary measures when the diffusion moves in some non convex environment (see [5]). The aim of this paper is to describe these invariant measures and especially their asymptotic behavior as the noise intensity in the nonlinear SDE becomes small. We prove in particular that the limit measures are discrete measures and point out some properties of their support which permit in several situations to describe explicitly the whole set of limit measures. This study requires essentially generalized Laplace's method approximations.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2087-2116

Publication Date: July 6, 2010

DOI: 10.1214/EJP.v15-842

References

  1. S. Benachour, B. Roynette, D. Talay, P. Vallois. Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75 (1998), no. 2, 173--201. Math. Review 99j:60079
  2. S. Benachour, B. Roynette, P. Vallois. Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stochastic Process. Appl. 75 (1998), no. 2, 203--224. Math. Review 99j:60080
  3. D.A. Dawson, J. G?rtner. Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987), no. 4, 247--308. Math. Review 89c:60092
  4. Funaki, Tadahisa. A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete 67 (1984), no. 3, 331--348. Math. Review 86b:35086
  5. S. Herrmann, J. Tugaut. Non-uniqueness of stationary measures for self-stabilizing processes. Stochastic Process. Appl. 120 (2010), 1215--1246. Math. Review number not available.
  6. H.P. McKean, Jr. A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1907--1911. Math. Review 36 #4647
  7. H.P. McKean, Jr. Propagation of chaos for a class of non-linear parabolic equations. 1967 Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) , 41--57 Air Force Office Sci. Res., Arlington, Va. Math. Review 38 #1759
  8. A.-S. Sznitman. Topics in propagation of chaos. ?cole d'?tÈ de ProbabilitÈs de Saint-Flour XIX---1989, 165--251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. Math. Review 93b:60179
  9. Y. Tamura. On asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 1, 195--221. Math. Review 86a:60133
  10. Y. Tamura. Free energy and the convergence of distributions of diffusion processes of McKean type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 443--484. Math. Review 89e:60151
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.