## Electronic Journal of Probability

### Large Deviations for One Dimensional Diffusions with a Strong Drift

Jochen Voss

#### Abstract

We derive a large deviation principle which describes the behaviour of a diffusion process with additive noise under the influence of a strong drift. Our main result is a large deviation theorem for the distribution of the end-point of a one-dimensional diffusion with drift $\theta b$ where $b$ is a drift function and $\theta$ a real number, when $\theta$ converges to $\infty$. It transpires that the problem is governed by a rate function which consists of two parts: one contribution comes from the Freidlin-Wentzell theorem whereas a second term reflects the cost for a Brownian motion to stay near a equilibrium point of the drift over long periods of time.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 53, 1479-1528.

Dates
Accepted: 1 September 2008
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819126

Digital Object Identifier
doi:10.1214/EJP.v13-564

Mathematical Reviews number (MathSciNet)
MR2438814

Zentralblatt MATH identifier
1193.60038

Subjects
Primary: 60F10 60H10

Rights

#### Citation

Voss, Jochen. Large Deviations for One Dimensional Diffusions with a Strong Drift. Electron. J. Probab. 13 (2008), paper no. 53, 1479--1528. doi:10.1214/EJP.v13-564. https://projecteuclid.org/euclid.ejp/1464819126

#### References

• Anderson, T. W. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, (1955). 170–176. (16,1005a)
• Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
• Birkhoff, Garrett; Rota, Gian-Carlo. Ordinary differential equations. Fourth edition. John Wiley & Sons, Inc., New York, 1989. xii+399 pp. ISBN: 0-471-86003-4
• Borodin, Andrei N.; Salminen, Paavo. Handbook of Brownian motion–-facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel, 1996. xiv+462 pp. ISBN: 3-7643-5463-1
• Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
• Gelfand, I. M.; Fomin, S. V. Calculus of variations. Revised English edition translated and edited by Richard A. Silverman Prentice-Hall, Inc., Englewood Cliffs, N.J. 1963 vii+232 pp.
• Voss, J. Some Large Deviation Results for Diffusion Processes. PhD thesis, University of Kaiserslautern, Germany, 2004.