From Markovian to non-Markovian persistence exponents
We present an exact formula relating the survival probability for certain Lévy flights (viz. asymmetric α-stable processes where ) 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, this 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.
Reference : EPL 109 40015