Persistence probabilities \& exponents

Abstract

This article deals with the asymptotic behaviour as $t\to +\infty$ of the survival function $P[T>t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behaviour is of the type $P[T>t]=t^{-\theta +o(1)}$ for a known or unknown positive parameter $\theta$ which is called a persistence exponent. The problem is well understood for random walks or Lévy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.

survey paper, submitted

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Cited by in corpus (11)

• First passage times of Lévy processes over a one-sided moving boundary
• Persistence probabilities for an integrated random walk bridge
• Persistence exponent for discrete-time, time-reversible processes
• Random walks and branching processes in correlated Gaussian environment
• Persistence probabilities for stationary increment processes
• Persistence of some additive functionals of Sinai's walk
• Exponential moments of first passage times and related quantities for Lévy processes
• The first passage time problem over a moving boundary for asymptotically stable Lévy processes
• Persistence of integrated stable processes
• Persistence exponent for random walk on directed versions of $Z^2$
• Universal Persistence for Local Time of One-dimensional Random Walk