Dichotomy in a scaling limit under Wiener measure with density

Tadahisa Funaki (Graduate School of Mathematical Sciences, The University of Tokyo)

Abstract


In general, if the large deviation principle holds for a sequence of probability measures and its rate functional admits a unique minimizer, then the measures asymptotically concentrate in its neighborhood so that the law of large numbers follows. This paper discusses the situation that the rate functional has two distinct minimizers, for a simple model described by the pinned Wiener measures with certain densities involving a scaling. We study their asymptotic behavior and determine to which minimizers they converge based on a more precise investigation than the large deviation's level.

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Pages: 173-183

Publication Date: May 16, 2007

DOI: 10.1214/ECP.v12-1271

References

  1. E. Bolthausen, T. Funaki and T. Otobe. Concentration under scaling limits for weakly pinned Gaussian random walks. preprint, 2007.
  2. T. Funaki. Stochastic Interface Models. Lectures on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint-Flour XXXIII - 2003 (ed. J. Picard), 103--274, Lect. Notes Math., 1869 (2005), Springer. MR2228384 (2007b:60236)
  3. T. Funaki. A scaling limit for weakly pinned Gaussian random walks. submitted to the Proceedings of RIMS Workshop on Stochastic Analysis and Applications, German-Japanese Symposium. RIMS Kokyuroku Bessatsu.
  4. T. Funaki and H. Sakagawa. Large deviations for Å?É” interface model and derivation of free boundary problems. Proceedings of Shonan/Kyoto meetings ";Stochastic Analysis on Large Scale Interacting Systems"; (2002, eds. Funaki and Osada). Adv. Stud. Pure Math., 39, Math. Soc. Japan, 2004, 173--211. MR2073334 (2006b:60218)
  5. I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus (2nd edition). Springer, 1991. MR1121940 (92h:60127)
  6. L. Takács. The distribution of the sojourn time for the Brownian excursion. Methodol. Comput. Appl. Probab., 1 (1999), 7--28. MR1714672 (2001a:60092)
  7. S.R.S. Varadhan. Large Deviations and Applications. SIAM, 1984. MR0758258 (86h:60067b)
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