Essential spectrum for Brox-type diffusion processes

This paper investigates the essential spectrum and compactness property of Markov semigroups generated by multi-dimensional Brox diffusion processes under two types of random media: stationary Gaussian random fields and semi-selfsimilar L\'evy random fields. For Gaussian environments satisfying general stationary covariance growth conditions, we prove that the associated semigroup is almost surely noncompact; if the covariance function grows sublinearly at infinity, then the bottom of the essential spectrum vanishes almost surely, with fractional Brownian fields as a concrete example. For multi-dimensional semi-selfsimilar L\'evy environments with scaling index $\alpha\in(1,2)$, we show that the essential spectral bottom equals zero almost surely for arbitrary space dimension $d\ge1$, and the same conclusion holds for one-dimensional symmetric $\alpha$-stable L\'evy processes with any $\alpha\in(0,2)$. Furthermore, we study one-dimensional diffusion operators perturbed by random L\'evy drift. When $0<\delta\le 1/\alpha$, random environmental fluctuations can destroy the compactness of semigroups induced by deterministic power-law potentials $\pm|x|^\delta$, even if the unperturbed semigroup is compact. The analysis relies on almost sure volume growth estimates for the random reference measure induced by environmental potentials, sample path asymptotics of random fields, ergodic theory of scaling transforms, and spectral criteria for regular Dirichlet forms. A unified framework linking sample path behaviors of random potentials to spectral characteristics of diffusion semigroups is established, and several open problems for large potential exponents are stated.