Fast rare events in exit times distributions of jump processes

Abstract

Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We formulate a general approach for estimating the contribution of fast rare events to the exit probabilities in the presence of fat tailed distributions. Using this approach, we study three jump processes that are used to model a wide class of phenomena ranging from biology to transport in disordered systems, ecology and finance: discrete time random-walks, Lévy walks and the Lévy-Lorentz gas. We determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits from an interval in a very short time at a large distance opposite to the starting point. In particular, we show that events occurring on time scales orders of magnitude smaller than the typical time scale of the process can make a significant contribution to the exit probability. Our results are confirmed by extensive numerical simulations.

arXiv comment6 pages, 4 figures

References in corpus (28)

• Intermittent search strategies
• Lévy walks
• Fluctuation-Dissipation: Response Theory in Statistical Physics
• Light fields in complex media: mesoscopic scattering meets wave control
• Optimal search strategies for hidden targets
• Distance matters: the impact of gene proximity in bacterial gene regulation
• The Single Big Jump Principle in Physical Modelling
• First detected arrival of a quantum walker on an infinite line
• One-Dimensional Stochastic Lévy-Lorentz Gas
• Lévy walks and scaling in quenched disordered media
• Absence of Normal Fluctuations in an Integrable Magnet
• Effective target arrangement in a deterministic scale-free graph
• Optimal non-Markovian search strategies with n-step memory
• Rare events in generalized Lévy Walks and the Big Jump principle
• Nonalgebraic length dependence of transmission through a chain of barriers with a Levy spacing distribution
• Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line
• Transport in disordered systems: the single big jump approach
• Splitting Probabilities of Jump Processes
• Extreme first passage times for random walks on networks
• Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
• Random walks in a one-dimensional Lévy random environment
• Rare events in stochastic processes with sub-exponential distributions and the Big Jump principle
• Scattering lengths and universality in superdiffusive Lévy materials
• Instantons for rare events in heavy-tailed distributions
• Non-homogeneous persistent random walks and averaged environment for the Lévy-Lorentz gas
• Continuous-time random walk between Lévy-spaced targets in the real line
• Large fluctuations and transport properties of the Lévy-Lorentz gas
• Leftward, Rightward and Complete Exit Time Distributions of Jump Processes

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• Mean first passage times of higher-dimensional velocity jump processes
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