Alberto Bassanoni (Parma)

Rare events and single big jump effects in stochastic processes

Rare events in stochastic processes are typically described within large deviation theory (LDT), where atypical fluctuations arise from the accumulation of many small contributions. In systems with sub-exponential statistics, however, rare events can instead be dominated by a single large fluctuation, as prescribed by the big jump principle (BJP). In this talk, I will discuss this alternative mechanism and its interplay with standard large deviation behavior across different classes of stochastic processes. I will first focus on power-law dynamics, such as Lévy processes, where single big jump effects control extreme value statistics and first-passage properties, including the behavior of the fastest trajectories in multi-particle settings, in particular their mean exit time from a bounded domain. I will then turn to processes with stretched-exponential statistics, with particular emphasis on the Ornstein–Uhlenbeck process. Using a renewal representation, one can identify a crossover between a regime of typical fluctuations described by LDT and a rare-event regime governed by the BJP, providing a physical interpretation of previously observed anomalous solutions in terms of single big jump effects. Finally, I will briefly discuss a perturbative approach that allows one to access intermediate regimes of moderate deviations, interpolating between these two limits.