From Markovian to non-Markovian persistence exponents
SAMM (EA 4543), Université Paris-1 Panthéon-Sorbonne, Centre Pierre Mendès-France - 90 rue de Tolbiac, 75013 Paris, France
Received: 30 December 2014
Accepted: 11 February 2015
We establish 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: . 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.
PACS: 05.40.Fb – Random walks and Levy flights / 05.40.Jc – Brownian motion / 02.50.Ey – Stochastic processes
© EPLA, 2015