Instantons for rare events in heavy-tailed distributions

Abstract

Large deviation theory and instanton calculus for stochastic systems are widely used to gain insight into the evolution and probability of rare events. At its core lies the realization that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by "convexifying" the rate function through nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation.

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• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems
• Eddy-Viscous Modeling and the Topology of Extreme Circulation Events in Three-Dimensional Turbulence
• Short-time large deviations of the spatially averaged height of a KPZ interface on a ring
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Extreme events and instantons in Lagrangian passive scalar turbulence models
• Instantons, fluctuations and singularities in the supercritical stochastic nonlinear Schroedinger equation