Electronic Journal of Probability

An invariance principle for Brownian motion in random scenery

Yu Gu and Guillaume Bal

Full-text: Open access


We prove an invariance principle for Brownian motion in Gaussian or Poissonian random scenery by the method of characteristic functions. Annealed asymptotic limits are derived in all dimensions, with a focus on the case of dimension $d=2$, which is the main new contribution of the paper.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 1, 19 pp.

Accepted: 2 January 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60F17: Functional limit theorems; invariance principles 60F05: Central limit and other weak theorems

weak convergence random media central limit theorem

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


Gu, Yu; Bal, Guillaume. An invariance principle for Brownian motion in random scenery. Electron. J. Probab. 19 (2014), paper no. 1, 19 pp. doi:10.1214/EJP.v19-2894. https://projecteuclid.org/euclid.ejp/1465065643

Export citation


  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • Bolthausen, Erwin. A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989), no. 1, 108–115.
  • Y. Gu and G. Bal, Weak convergence approach to a parabolic equation with large, highly oscillatory, random potential, preprint arXiv:1304.5005, (2013).
  • Kesten, H.; Spitzer, F. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 5–25.
  • Kipnis, C.; Varadhan, S. R. S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986), no. 1, 1–19.
  • Komorowski, Tomasz; Landim, Claudio; Olla, Stefano. Fluctuations in Markov processes. Time symmetry and martingale approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 345. Springer, Heidelberg, 2012. xviii+491 pp. ISBN: 978-3-642-29879-0
  • Lejay, Antoine. Homogenization of divergence-form operators with lower-order terms in random media. Probab. Theory Related Fields 120 (2001), no. 2, 255–276.
  • Pardoux, Etienne; Piatnitski, Andrey. Homogenization of a singular random one dimensional PDE. Multi scale problems and asymptotic analysis, 291–303, GAKUTO Internat. Ser. Math. Sci. Appl., 24, Gakkōtosho, Tokyo, 2006.
  • Rémillard, Bruno; Dawson, Donald A. A limit theorem for Brownian motion in a random scenery. Canad. Math. Bull. 34 (1991), no. 3, 385–391.
  • Sznitman, Alain-Sol. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+353 pp. ISBN: 3-540-64554-3.