The Annals of Probability

Moderate deviations for diffusions with Brownian potentials

Yueyun Hu and Zhan Shi

Full-text: Open access

Abstract

We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani’s lemma and Lamperti’s representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571–609] who studied the corresponding problem for Sinai’s random walk in random environment.

Article information

Source
Ann. Probab., Volume 32, Number 4 (2004), 3191-3220.

Dates
First available in Project Euclid: 8 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1107883351

Digital Object Identifier
doi:10.1214/009117904000000829

Mathematical Reviews number (MathSciNet)
MR2094443

Zentralblatt MATH identifier
1066.60096

Subjects
Primary: 60K37: Processes in random environments 60F10: Large deviations

Keywords
Moderate deviation diffusion with random potential Brownian valley

Citation

Hu, Yueyun; Shi, Zhan. Moderate deviations for diffusions with Brownian potentials. Ann. Probab. 32 (2004), no. 4, 3191--3220. doi:10.1214/009117904000000829. https://projecteuclid.org/euclid.aop/1107883351


Export citation

References

  • Bass, R. F. and Griffin, P. S. (1985). The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete 70 417--436.
  • Brox, T. (1986). A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14 1206--1218.
  • Comets, F. and Popov, S. (2003). Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Related Fields 126 571--609.
  • Csáki, E. and Földes, A. (1987). A note on the stability of the local time of a Wiener process. Stochastic Process. Appl. 25 203--213.
  • Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2, 2nd ed. Wiley, New York.
  • Gantert, N. and Zeitouni, O. (1999). Large deviations for one-dimensional random walk in a random environment---a survey. In Random Walks (P. Révész and B. Tóth, eds.) 127--165. Bolyai Math. Soc., Budapest.
  • Getoor, R. K. and Sharpe, M. J. (1979). Excursions of Brownian motion and Bessel processes. Z. Wahrsch. Verw. Gebiete 47 83--106.
  • Hu, Y. and Shi, Z. (1998). The limits of Sinai's simple random walk in random environment. Ann. Probab. 26 1477--1521.
  • Hu, Y., Shi, Z. and Yor, M. (1999). Rates of convergence of diffusions with drifted Brownian potentials. Trans. Amer. Math. Soc. 351 3915--3934.
  • Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
  • Kawazu, K. and Tanaka, H. (1997). A diffusion process in a Brownian environment with drift. J. Math. Soc. Japan 49 189--211.
  • Kent, J. (1978). Some probabilistic properties of Bessel functions. Ann. Probab. 6 760--770.
  • Kesten, H. (1965). An iterated logarithm law for local time. Duke Math. J. 32 447--456.
  • Lamperti, J. (1972). Semi-stable Markov processes, I. Z. Wahrsch. Verw. Gebiete 22 205--225.
  • Le Doussal, P., Monthus, C. and Fisher, D. S. (1999). Random walkers in one-dimensional random environments: Exact renormalization group analysis. Phys. Rev. E 59 4795--4840.
  • Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 425--457.
  • Révész, P. (1990). Random Walk in Random and Non-Random Environments. World Scientific, Singapore.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • Schumacher, S. (1985). Diffusions with random coefficients. Contemp. Math. 41 351--356.
  • Shi, Z. (2001). Sinai's walk via stochastic calculus. In Panoramas et Synthèses 12 (F. Comets and E. Pardoux, eds.). Société Mathématique de France.
  • Sinai, Ya. G. (1982). The limiting behaviours of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 256--268.
  • Taleb, M. (2001). Large deviations for a Brownian motion in a drifted Brownian potential. Ann. Probab. 29 1173--1204.
  • Tanaka, H. (1995). Diffusion processes in random environments. In Proc. of the International Congress of Mathematicians 2 (S. D. Chatterji, ed.) 1047--1054. Birkhäuser, Basel.
  • Warren, J. and Yor, M. (1998). The Brownian burglar: Conditioning Brownian motion by its local time process. Séminaire de Probabilités XXXII. Lecture Notes in Math. 1686 328--342. Springer, Berlin.
  • Zeitouni, O. (2001). Lecture notes on random walks in random environment. Saint Flour Lecture Notes.