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

TL;DR

This paper derives a variational formula for the Lyapunov exponent of Brownian motion in a stationary ergodic potential, extending previous results to non-compact settings and providing insights into the effects of randomness on decay rates.

Contribution

It generalizes Schroeder's variational formula to non-compact potentials and establishes exponential decay of the Green function for Brownian motion in random environments.

Findings

Derived a variational formula for the Lyapunov exponent

Established exponential decay of the Green function

Provided a variational expression for quenched free energy

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.