  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.
