Anomalous diffusion in presence of a (fractal) boundary is investigated. Asymptotic behaviour of the survival probability and current through an absorbing boundary are exactly calculated in specific examples. They agree with recent findings related with anomalous Warburg impedance experiments. The autocorrelation function for sites near the boundary is carefully discussed ; in presence of reflecting boundary conditions it can be different from the (spatial) average of the autocorrelation at variance with the normal diffusion case. Interesting issues from a methodological point of view are discussed in the renormalization groups analysis. Scaling arguments are given relating various exponents.
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