Electronic Communications in Probability

Dichotomy in a scaling limit under Wiener measure with density

Tadahisa Funaki

Full-text: Open access


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.

Article information

Electron. Commun. Probab., Volume 12 (2007), paper no. 18, 173-183.

Accepted: 16 May 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 82B24: Interface problems; diffusion-limited aggregation 82B31: Stochastic methods

Large deviation principle minimizers pinned Wiener measure scaling limit concentration

This work is licensed under aCreative Commons Attribution 3.0 License.


Funaki, Tadahisa. Dichotomy in a scaling limit under Wiener measure with density. Electron. Commun. Probab. 12 (2007), paper no. 18, 173--183. doi:10.1214/ECP.v12-1271. https://projecteuclid.org/euclid.ecp/1465224961

Export citation


  • E. Bolthausen, T. Funaki and T. Otobe. Concentration under scaling limits for weakly pinned Gaussian random walks. preprint, 2007.
  • 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.
  • 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.
  • T. Funaki and H. Sakagawa. Large deviations for $\nabla_\varphi$ interface model and derivation of free boundary problems. Proceedings of Shonan/Kyoto meetings ";Stochastic Analysis on Large Scale Interacting Systems";. Adv. Stud. Pure Math., 39, Math. Soc. Japan, 2004, 173–211.
  • I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus (2nd edition). Springer, 1991.
  • L. Takács. The distribution of the sojourn time for the Brownian excursion. Methodol. Comput. Appl. Probab., 1 (1999), 7–28.
  • S.R.S. Varadhan. Large Deviations and Applications. SIAM, 1984.