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

TL;DR

This paper establishes bounds on the expected main eigenvalue and quenched pressure of the Ruelle operator for Brownian motion-derived potentials, using a novel isomorphism with the Skorokhod space.

Contribution

It introduces new bounds for the eigenvalue and pressure in a stochastic setting and constructs a stochastic functional equation via a novel isomorphism.

Findings

Derived bounds for the expected eigenvalue and pressure.

Established an isomorphism with the Skorokhod space.

Formulated a stochastic functional equation for eigenvalues.

Abstract

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.