Rare events in generalized Lévy Walks and the Big Jump principle

Abstract

The prediction and estimate of rare events is an important task in disciplines that range from physics and biology, to economics and social science. A peculiar aspect of the mechanism that drives rare events is described by the so called Big Jump Principle. According to the principle, in heavytailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic models with wide applications, from complex transport to search processes, that accounts for complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution feature non-universal and non-analytic behaviors, that crucially depend on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.

Big jump principle for heavy-tailed random walks with correlated increments

“…The latter assumes fixed b, see Eq. (11). For large N , there is a non-negligible amount of W N −b which is not similar to this limit.…”

“…Here, it is important that we don't fix b as previously in Eq. ( 10) and (11). We explain now both cases in detail.…”

“…The main challenge is the corporation of correlations into the theory. Recently, the BJP draw attention in statistical physics [8][9][10][11][12]. These works studied, amongst other things, Lévy walks, renewal processes, continuous time random walks and the Lévy Lorentz gas.…”

Diffusion and escape from polygonal channels: extreme values and geometric effects

“…However, the fraction of such fastest trajectories does necessarily decrease in time, since it approaches the set of exactly horizontal trajectories. With infinite horizon it would be possible in principle to describe the transport in polygonal channels in terms of the single big jump principle [43,54]. However, our results suggest that the sub leading ballistic contribution vanishes rapidly enough to affect the properties of transport.…”

Splitting Probabilities of Jump Processes

“…Levy Flights. For jump processes with infinite second moment ie µ < 2 in (2) -called Levy flights [17,18,30,31], no exact results for the splitting probability are available for generic a µ , x 0 . We thus resort to numerical simulations to validate predictions (11) to (14) (see Fig.…”