Random environmental potentials (Gaussian or semi-selfsimilar Lévy) almost surely yield noncompact Markov semigroups with zero essential spectral bottom, destroying deterministic confinement.
Orey, Growth rate of certain Gaussian processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ
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Essential spectrum for Brox-type diffusion processes
Random environmental potentials (Gaussian or semi-selfsimilar Lévy) almost surely yield noncompact Markov semigroups with zero essential spectral bottom, destroying deterministic confinement.