  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.
