39 research outputs found
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