|
Résumé |
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. |
