Second order asymptotics for Brownian motion in a heavy tailed Poissonian potential

We consider the Feynman-Kac functional associated with a Brownian motion in a random potential. The potential is defined by attaching a heavy tailed positive potential around the Poisson point process. This model was first considered by Pastur (1977) and the first order term of the moment asymptotics was determined. In this paper, both moment and almost sure asymptotics are determined up to the second order. As an application, we also derive the second order asymptotics of the integrated density of states of the corresponding random Schr\"odinger operator.Comment: 29 pages. Minor correction

Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.Comment: 27 page

Annealed Brownian motion in a heavy tailed Poissonian potential

Consider a d-dimensional Brownian motion in a random potential defined by attaching a nonnegative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson points by weighting the measure by the Feynman-Kac functional. We show that under the weighted measure, the Brownian motion tends to localize around the origin. We also determine the scaling limit of the path and also the limit shape of the random potential.Comment: Published in at http://dx.doi.org/10.1214/12-AOP754 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Slowdown estimates for one-dimensional random walks in random environment with holding times

We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.Comment: 13 pages. There are corrections in the extreme value lemmas and the quenched slowdown estimate