From Markovian to non-Markovian persistence exponents

Abstract

We establish an exact formula relating the survival probability for certain L\'evy flights (viz. asymmetric $\alpha$-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 $\delta$ in the latter, non Markovian case is simply related to the persistence exponent $\theta$ 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.