From Markovian to non-Markovian persistence exponents

We establish an exact formula relating the survival probability for certain Lévy flights (viz. asymmetric $α$-stable processes where $α= 1/2$) with the survival probability for the order statistics of the running maxima of two independent Brownian particles. This formula allows us to show that the persistence exponent $δ$ in the latter, non Markovian case is simply related to the persistence exponent $θ$ in the former, Markovian case via: $δ=θ/2$. Thus, our formula reveals a link between two recently explored families of anomalous exponents: one exhibiting continuous deviations from Sparre-Andersen universality in a Markovian context, and one describing the slow kinetics of the non Markovian process corresponding to the difference between two independent Brownian maxima.

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