From Markovian to non-Markovian persistence exponents

Abstract

We establish an exact formula relating the survival probability for certain Lévy flights (viz. asymmetric α-stable processes where α = 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 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, 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.

Maxima of two random walks: universal statistics of lead changes

“…This distribution would imply that the probability of having no lead change decays as exp[−(ln t)/π] ∼ t −1/π , that is, slower than the f 0 ∼ t −1/4 behavior which has been established analytically [13,14].…”

“…This distribution would imply that the probability of having no lead change decays as exp[−(ln t)/π] ∼ t −1/π , that is, slower than the f 0 ∼ t −1/4 behavior which has been established analytically [13,14]. It is straightforward to generalize equation (4) to the situation where the two random walks have different diffusion coefficients, denoted by D 1 and D 2 .…”

“…We now focus on the probability f n (t) to have exactly n lead changes until time t. For two identical random walks, the probability that there are no lead changes decays as f 0 ∼ t −1/4 in the long-time limit [13,14]. Since f n (t) is the probability that the (n + 1)-st lead change takes place after time t, the probability density for the (n + 1)-st lead change to occur at time t is given by −df n /dt.…”

“…1). The probability that one maximum exceeds another during the time interval (0, t) was investigated only recently [13,14]. Here we explore additional features of the interplay between two Brownian maxima.…”

Scaling exponents for ordered maxima

“…This investigation is motivated by a recent letter [26] concerning maximal positions of random walks. It was reported that the probability that the maxima of multiple random walks remain perfectly ordered decays as a power law with the number of steps [26,27], and that the corresponding decay exponents are generally nontrivial.…”