Raffaella Burioni

Dipartimento di Fisica, Università di Parma, Italy

Large Fluctuations and Fast Rare events in First Passage probabilities

A stochastic process that reaches a certain threshold for the first time can trigger many events: a chemical reaction occurs, a target is reached, biological and ecological processes take place. The study of these triggering events and the probability of their occurrence is based on the knowledge of first-passage probabilities. Because of their "triggering" property, the tails of these distributions are particularly important as they allow the probability of rare anomalous events to be estimated, such as leaving an interval in a very short time. A few results are available for the first-pass probabilities of jump processes, which are stochastic processes that involve random jumps between different states or positions, occurring at random times and with random magnitudes. When jump distributions are heavy-tailed, the mechanism that leads to rare events is peculiar. Instead of being caused by a set of many small deviations all in the same direction, one jump, the largest of the lot, provides the main contribution to the rare large fluctuation.  We will review some recent results that allow the study of rare events and fast exit times from an interval in a broad class of Lévy processes, thanks to the ‘big jump’ approach.