Alberto Bassanoni — University of Parma #
Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes #
We study the full probability distribution of the time-integrated velocity of a particle elevated to n-th power, with n greater than two, where the velocity follows the Ornstein-Uhlenbeck dynamics. Using the standard tools of renewal theory, we treat the problem as a decoupled continuous time random walk. The probability distribution of our initial observable is related to the problem of the area under the Ornstein-Uhlenbeck excursions, which we treat with the usual methods of first passage problems. By using a formalism of large deviations, we derive the full probability distribution of the first passage area under the excursions, and we find that it has two different phases, directly depending by the parameters of the initial process: a diffusive Brownian phase in the strong noise limit and an anomalous sub-exponential phase in the weak noise limit, with a dynamical phase transition at the deterministic limit. Once we get the waiting times and the areas statistics, we are able to reconstruct our desired probability distribution through the continuous time random walk mapping. After calculating its typical Gaussian fluctuations, the main focus of our work is on studying its rare events distribution using the single big jump principle. We discover in the study of the tail process that the single big jump formalism returns the same results of other works obtained by the optimal functional method, a path integral approach that identifies the source of the anomalous subexponential scaling with the presence of an instantonic solution in the weak noise regime. This formal analogy will be discussed and tested with extensive numerical simulations, showing that the single big jump principle is more general, and this opens the way to further extensions of our theory in the study of rare events of a broader class of Langevin processes;