Analytical results for random walk persistence
Résumé
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 $X_0\\sqrt{}$.