3 research outputs found
Continuity Results and Estimates for the Lyapunov Exponent of Brownian Motion in Random Potential
We collect some applications of the variational formula established by
Schr\"oder (1988) and Rue\ss (2013) for the quenched Lyapunov exponent of
Brownian motion in stationary and ergodic nonnegative potential. We show for
example that the Lyapunov exponent for nondeterministic potential is strictly
lower than the Lyapunov exponent for the averaged potential. The behaviour of
the Lyapunov exponent under independent perturbations of the underlying
potential is examined. And with the help of counterexamples we are able to give
a detailed picture of the continuity properties of the Lyapunov exponent.Comment: 20 pages, 1 figure, some references update
Lyapunov Exponents of Brownian Motion: Decay Rates for Scaled Poissonian Potentials and Bounds
We investigate Lyapunov exponents of Brownian motion in a nonnegative
Poissonian potential . The Lyapunov exponent depends on the potential
and our interest lies in the decay rate of the Lyapunov exponent if the
potential tends to zero. In our model the random potential is generated
by locating at each point of a Poisson point process with intensity a
bounded compactly supported nonnegative function . We show that for
sequences of potentials for which for some
constant (), the decay rates to zero of the quenched and
annealed Lyapunov exponents coincide and equal where the constant
is computed explicitly. Further we are able to estimate the quenched
Lyapunov exponent norm from above by the corresponding norm for the averaged
potential.Comment: Now 14 pages, 2 figures. Some references added, abstract changed, 2
new paragraphs in the introductio
A Variational Formula for the Lyapunov Exponent of Brownian Motion in Stationary Ergodic Potential
We establish a variational formula for the exponential decay rate of the
Green function of Brownian motion evolving in a random stationary and ergodic
nonnegative potential. Such a variational formula is established by Schroeder
in 'Green's Functions for the Schr\"odinger Operator with Periodic Potential',
J. Funct. Anal. 77 (1988), for potentials on compact spaces and is generalised
in the present article to a non-compact setting. We show exponential decay of
the Green function implicitly. This formula for the Lyapunov exponent has
several direct implications. It allows to compare the influence of a random
potential to the influence of the averaged potential. It also leads to a
variational expression for the quenched free energy.Comment: 26 pages, minor correction