Annealed deviations of random walk in random scenery |
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Authors: | Nina Gantert Zhan Shi |
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Institution: | a Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany b Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany c Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4, place Jussieu, 75252 Paris Cedex 05, France |
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Abstract: | Let (Zn)n∈N be a d-dimensional random walk in random scenery, i.e., with (Sk)k∈N0 a random walk in Zd and (Y(z))z∈Zd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of for various choices of sequences n(bn) in 1,∞). Depending on n(bn) and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497-527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004]. |
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Keywords: | 60K37 60F10 60J55 |
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