Fractional processes conditioned to stay positive

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We study fractional Brownian motion and related fractional processes. The goal is to define fractional Brownian motion conditioned to stay positive.The difficulty comes from the fact that the event we condition on -- staying positive forever -- has probability zero. Additionally, the considered process is intrinsically non-Markovian.This new object generalizes 'Brownian motion conditioned to stay positive'. Extensions to other fractional processes are also investigated; methods come from the theory of persistence probabilities studied intensively over the last 5-7 years.