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We establish a variational formula for the exponential decay rate of theGreen function of Brownian motion evolving in a random stationary and ergodicnonnegative potential. Such a variational formula is established by Schroederin 'Green's Functions for the Schr\"odinger Operator with Periodic Potential',J. Funct. Anal. 77 (1988), for potentials on compact spaces and is generalisedin the present article to a non-compact setting. We show exponential decay ofthe Green function implicitly. This formula for the Lyapunov exponent hasseveral direct implications. It allows to compare the influence of a randompotential to the influence of the averaged potential. It also leads to avariational expression for the quenched free energy.

A Variational Formula for the Lyapunov Exponent of Brownian Motion in Stationary Ergodic Potential