Instantons for rare events in heavytailed 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 fact that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddlepoint 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 nonconvex large deviation rate function. We propose a solution to this problem by 'convexifying' the rate function through a nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of superexponential 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 MonteCarlo 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 April 2021
 DOI:
 10.1088/17518121/abe67b
 arXiv:
 arXiv:2012.03360
 Bibcode:
 2021JPhA...54q5001A
 Keywords:

 large deviation principle;
 exponentially tilted measures;
 nonconvex rate functions;
 nonlinear reparametrizations;
 instanton equations;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Mathematics  Optimization and Control
 EPrint:
 doi:10.1088/17518121/abe67b