Persistence probabilities \& exponents

TL;DR

This paper surveys the asymptotic behavior of persistence probabilities and exponents for stochastic processes, focusing on the decay rate of survival functions and highlighting recent advances and open problems in the field.

Contribution

It provides a comprehensive overview of recent results and open questions regarding persistence probabilities and exponents for various stochastic processes, especially integrals of Lévy processes.

Findings

Behavior characterized by power-law decay with persistence exponent θ

Well-understood for random walks and Lévy processes

More complex for integrals of such processes, with ongoing research

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 L\'evy 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.