  In this paper we consider a random potential derived from the Brownian
motion. We obtain upper and lower bounds for the expected value of the main
eigenvalue of the associated Ruelle operator and for its quenched topological
pressure. We also exhibit an isomorphism between the space $C(\Omega)$ endowed
with its standard norm and a proper closed subspace of the Skorokhod space
which is used to obtain a stochastic functional equation for the main
eigenvalue and for its associated eigenfunction.
