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In this paper we consider a random potential derived from the Brownianmotion. We obtain upper and lower bounds for the expected value of the maineigenvalue of the associated Ruelle operator and for its quenched topologicalpressure. We also exhibit an isomorphism between the space $C(\Omega)$ endowedwith its standard norm and a proper closed subspace of the Skorokhod spacewhich is used to obtain a stochastic functional equation for the maineigenvalue and for its associated eigenfunction.

Bounds on the Quenched Pressure and Main Eigenvalue of the Ruelle Operator for Brownian Type Potentials