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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.

Continuity results and estimates for the Lyapunov exponent of Brownian motion in stationary potential