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Register a new userOne-dimensional space-discrete transport subject to Levy perturbations
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In this paper we study a one-dimensional space-discrete transport equation subject to additive Levy forcing. The explicit form of the solutions allows their analytic study. In particular we discuss the invariance of the covariance structure of the stationary distribution for Levy perturbations with finite second moment. The situation of more general Levy perturbations lacking the second moment is considered as well. We moreover show that some of the properties of the solutions are pertinent to a discrete system and are not reproduced by its continuous analogue.
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We consider a one-dimensional Anderson model where the potential decays in average like $n^{-alpha}$, $alpha>0$. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate $alpha$ varies. We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case $alpha>frac12$, a transition from singular continuous to pure point spectrum in the critical case $alpha=frac12$, and dense pure point spectrum in the sub-critical case $alpha<frac12$. We present complete proofs of the cases $alphagefrac12$ and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for $alpha=frac12$. Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region $alpha<frac12$. This implies, in particular, that the spectrum is pure point in this regime.
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