Rare events and single big jump effects in Ornstein–Uhlenbeck processes

Abstract Even in a simple stochastic process, the study of the full distribution of time-integrated observables can be a difficult task. This is the case of a much-studied process such as the Ornstein–Uhlenbeck process where, recently, anomalous dynamical scaling of large deviations of time-integrated functionals has been highlighted. Using the mapping of a continuous stochastic process to a continuous time random walk via the ‘excursions technique’, we introduce a comprehensive formalism that enables the calculation of the complete distribution of the time-integrated observable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>v</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> , where n is a positive integer and v ( t ) is the random velocity of a particle following Ornstein–Uhlenbeck dynamics. We reveal an interesting connection between the anomalous rate function associated with the observable A and the statistics of the area under the first-passage functional during an excursion. The rate function of the latter, analyzed here for the first time, exhibits anomalous scaling behavior and a critical point in its dynamics, both of which are explored in detail. The case of the anomalous scaling of large deviations, originally associated with the presence of an instantonic solution in the weak noise regime of a path integral approach, is here produced by a so-called ‘big jump effect’, in which the contribution to rare events is dominated by the largest excursion. Our approach, which is quite general for continuous stochastic processes, allows us to associate a physical meaning with the anomalous scaling of large deviations through the big jump principle.