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 $\alpha = 1/2$ ) 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: $\delta=\theta/2$ . 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