×

zbMATH — the first resource for mathematics

Annealed Lyapounov exponents and large deviations in a Poissonian potential. I. (English) Zbl 0826.60018
Author’s abstract: We study annealed Brownian motion moving in a Poissonian cloud of killing spheres of fixed radius (hard obstacles) or in a Poissonian potential (soft obstacles). “Annealed” refers to the fact that statistical weights of interest are averaged both with respect to the path and environment measures. We construct Lyapunov exponents which for instance in the soft obstacle case measure the directional exponential decay of the environment averaged Green’s function. These exponents come naturally in the description of certain large deviation principles which govern the large time position of annealed Brownian motion in a Poissonian potential, as well as in certain large time asymptotics of the associated heat kernel.
[For part II see below].

MSC:
60F10 Large deviations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] E. BOLTHAUSEN E. , A note on the diffusion of directed polymers in a random environment (Comm. Math. Phys., Vol. 123, 1989 , pp. 529-534). Article | MR 91a:60270 | Zbl 0684.60013 · Zbl 0684.60013 · doi:10.1007/BF01218584 · minidml.mathdoc.fr
[2] E. B. DAVIES , Analysis on graphs and non commutative geometry (J. Funct. Anal., Vol. 111, 1993 , pp. 398-430). MR 93m:58110 | Zbl 0793.46043 · Zbl 0793.46043 · doi:10.1006/jfan.1993.1019
[3] J. D. DEUSCHEL and D. W. STROOCK , Large Deviations (Academic Press, Boston, 1989 ). MR 90h:60026 | Zbl 0705.60029 · Zbl 0705.60029
[4] M. D. DONSKER and S. R. S. VARADHAN , Asymptotics for the Wiener sausage (Comm. Pure Appl. Math., Vol. 28, 1975 , pp. 525-565). MR 53 #1757a | Zbl 0333.60077 · Zbl 0333.60077 · doi:10.1002/cpa.3160280406
[5] H. KESTEN , Aspects of first passage percolation , Ecole d’été de Probabilités de St Flour (Lect. Notes in Math., Vol. 1180, 1986 , pp. 125-264, Springer, Berlin). MR 88h:60201 | Zbl 0602.60098 · Zbl 0602.60098
[6] P. LI and S. T. YAU , On the parabolic kernel of the Schrödinger operator (Acta Mathematica, Vol. 156, 1986 , pp. 153-201). MR 87f:58156 | Zbl 0611.58045 · Zbl 0611.58045 · doi:10.1007/BF02399203
[7] B. SIMON , Schrödinger semigroups (Bull. Am. Math. Soc., Vol. 7, 1986 , pp. 447-526). Article | MR 86b:81001a | Zbl 0524.35002 · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8 · minidml.mathdoc.fr
[8] A. S. SZNITMAN , On long excursions of Brownian motion among Poissonian obstacles in Stochastic Analysis (M. Barlow, N. Brigham, ed., London Math. Soc., Lecture Note Series, Cambridge University Press, 1991 , pp. 353-375). MR 93e:60167 | Zbl 0760.60027 · Zbl 0760.60027
[9] A. S. SZNITMAN , Brownian motion with a drift in a Poissonian potential (Comm. Pure Appl. Math., Vol. 47, 10, 1994 , pp. 1283-1318). MR 95m:60122 | Zbl 0814.60021 · Zbl 0814.60021 · doi:10.1002/cpa.3160471002
[10] A. S. SZNITMAN , Shape theorem, Lyapounov exponents and large deviations for Brownian motion in a Poissonian potential (Comm. Pure Appl. Math., Vol. 47, 12, 1994 , pp. 1655-1688). MR 96b:60217 | Zbl 0814.60022 · Zbl 0814.60022 · doi:10.1002/cpa.3160471205
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.