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 $V$. The Lyapunov exponent depends on the potential $V$ and our interest lies in the decay rate of the Lyapunov exponent if the potential $V$ tends to zero. In our model the random potential $V$ is generated by locating at each point of a Poisson point process with intensity $\nu$ a bounded compactly supported nonnegative function $W$. We show that for sequences of potentials $V_n$ for which $\nu_n \|W_n\|_1 \sim D/n$ for some constant $D > 0$ ($n \to \infty$), the decay rates to zero of the quenched and annealed Lyapunov exponents coincide and equal $c n^{-1/2}$ where the constant $c$ 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