×

zbMATH — the first resource for mathematics

Shape theorem, Lyapounov exponents, and large deviations for Brownian motion in a Poissonian potential. (English) Zbl 0814.60022
Author’s abstract: We derive a large deviation principle governing the position of a \(d\)-dimensional Brownian motion moving in a Poissonian potential. The derivation of this large deviation principle, and the form of the rate function rely on a result similar to the “shape theorem” of first passage percolation. This result produces certain constants which play in this multidimensional situation a similar role as the Lyapunov exponents in the one-dimensional case. The large deviation principle enables us to investigate the transition of regime, which occurs between the small \(| h |\) and the large \(| h | \) case, for Brownian motion with a constant drift \(h\) moving in the same potential.

MSC:
60F10 Large deviations
60J65 Brownian motion
PDF BibTeX Cite
Full Text: DOI
References:
[1] Boivin, Probab. Theory Related Fields 86 pp 491– (1990)
[2] and , Spectral Theory of Random Schrödinger Operators, Probability and its Applications, Birkhäuser, Basel, 1991.
[3] Lectures From Markov Processes to Brownian Motion, Springer-Verlag, New York, 1982. · Zbl 0503.60073
[4] Cox, Ann. Probab. 9 pp 583– (1981)
[5] and , Asymptotic evaluation of certain Wiener integrals for large time, pp. 15–33 in: Functional Integration and its Applications, ed., Proceedings of International Conference London, April 1974, Clarendon, Oxford 1975.
[6] Donsker, Comm. Pure Appl. Math. 28 pp 525– (1975)
[7] Entropy, Large Deviations, and Statistical Mechanics, Springer-Verlag, New York, 1985.
[8] Functional Integration and Partial Differential Equations, Annals of Mathematics Studies 109, Princeton University Press, 1985. · Zbl 0568.60057
[9] and , First passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, pp. 61–110 in: Bernoulli, Bayes, Laplace, Anniversary Volume, and , eds., Springer-Verlag, Berlin, 1965.
[10] Aspects of first passage percolation, pp. 125–264 in: Ecole d’été de Probabilités de St. Flour, Lecture Notes in Math. 1180, Springer-Verlag, Berlin, 1986.
[11] Interacting Particle Systems, Springer-Verlag, New-York, 1985.
[12] Real and Complex Analysis, second edition, McGraw-Hill, New York, 1974.
[13] Sznitman, Ann. Probab. 21 pp 490– (1993)
[14] Sznitman, Probab. Theory Related Fields 95 pp 155– (1993)
[15] Sznitman, Comm. Pure Appl. Math. 47 pp 1283– (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.