The infinitesimal generator of the Brox diffusion

TL;DR

This paper constructs the infinitesimal generator of the Brox diffusion in a periodic environment, providing new insights into its properties, ergodicity, and boundary behaviors.

Contribution

It introduces a novel construction of the Brox diffusion's generator, enabling analysis of its semigroup, boundary conditions, and ergodic properties.

Findings

Semigroup is strong Feller with Gaussian bounds

Existence of a unique invariant measure in bounded space

Spectral gap implies exponential ergodicity

Abstract

We construct the infinitesimal generator of the Brox diffusion on a line with a periodic Brownian environment. This gives a new construction of the process and allows to solve the singular martingale problem. We prove that the associated semigroup is strong Feller with Gaussian lower and upper bounds. This also yields a construction of the Brox diffusion on a segment with periodic or Dirichlet boundary conditions. In this bounded space, we prove that there exists a unique measure and the existence of a spectral gap giving exponential ergodicity of the diffusion.