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

Abstract

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.