The killed Brox diffusion

Abstract

We are proposing to study a diffusion in random environment confined in bounded interval. As for many standard diffusions, we build in natural way a stochastic process with bounded domain, but with addition of considering a random environment. We carry out this construction using the Brox diffusion and applying well-known diffusion theory in a quenched fashion, which is a natural way to deal with random environment. The outcome of this procedure is an object that we may call the killed Brox diffusion. Since the generator of this process is initially an ill-posed expression we develop a Sturm–Liouville theory for one-dimensional second-order differential operators with white-noise coefficients. Our first main result is to give a close form of the Green operator associated to the generator, i.e., the inverse of the generator. We do so by setting the Lagrange identity in this context. Then, we give explicit expressions in quenched form of the probability density function of the process; such an object is given theoretically in terms of the spectral decomposition using the eigenvalues and eigenfuntions of the infinitesimal generator of the diffusion. Moreover, we characterize the eigenvalues and eigenfunctions using some integro-differential equations.