Persistence probabilities \& exponents

Details Persistence probabilities \& exponents Persistence probabilities \& exponents

2012-03-29

arxiv.org/abs/1203.6554v1

Mathematics

Probability

survey paper, submitted

Scientific paper

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.

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