From Markovian to non-Markovian persistence exponents

TL;DR

This paper derives an exact formula linking the survival probabilities of certain Markovian and non-Markovian processes, revealing a simple relationship between their persistence exponents and connecting two families of anomalous exponents.

Contribution

The paper introduces an exact formula connecting survival probabilities of asymmetric Lévy flights and Brownian maxima, establishing a direct relation between their persistence exponents.

Findings

Persistence exponent in non-Markovian case is half of the Markovian case.

The formula links two families of anomalous exponents.

Reveals a universal relation between different stochastic processes.

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.