From Markovian to non-Markovian persistence exponents

Abstract

We establish an exact formula relating the survival probability for certain Lévy 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.

Accepted in EPL

References in corpus (7)

• Leapover lengths and first passage time statistics for Lévy flights
• Universal Record Statistics of Random Walks and Lévy Flights
• Persistence Problem in Two-Dimensional Fluid Turbulence
• Probability distribution of the maximum of a smooth temporal signal
• Slow Kinetics of Brownian Maxima
• Persistence of Random Walk Records
• Unusual eigenvalue spectrum and relaxation in the Lévy Ornstein-Uhlenbeck process

Cited by in corpus (1)

• Statistical properties of sites visited by independent random walks