Analytical results for random walk persistence

TL;DR

This paper provides detailed analytical calculations of the persistence exponent for Gaussian and non-Gaussian processes, revealing new formulas and connections to quantum mechanics, enhancing understanding of persistence probabilities.

Contribution

It introduces new resummed and non-perturbative expressions for the persistence exponent, extending the theory to non-Gaussian processes and linking it to quantum energy eigenfunctions.

Findings

Derived new formulas for the persistence exponent $ heta$.

Extended perturbation theory to non-Gaussian processes.

Connected persistence problems to quantum mechanical eigenfunctions.

Abstract

In this paper, we present the detailed calculation of the persistence exponent $\theta$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $\theta$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $\theta$ for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and non-perturbative expressions for the persistence exponent $\theta(X_0)$, describing the probability that the process remains bigger than $X_0\sqrt{<X^2(t)>}$.