Extreme Analysis of a Non-convex and Nonlinear Functional of Gaussian Processes -- On the Tail Asymptotics of Random Ordinary Differential Equations

Abstract

In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic deformation, water flow, electric-magnetic fields, temperature distribution, etc. A random coefficient is introduced to account for the system's uncertainty and/or imperfect measurements. This random coefficient is described by a Gaussian process (the input process) and thus the solution to the differential equation (under certain boundary conditions) is a complexed functional of the input Gaussian process. In this paper, we focus the analysis on the one-dimensional case and derive asymptotic approximations of the tail probabilities of the solution to the equation that has various physics interpretations under different contexts. This analysis rests on the literature of the extreme analysis of Gaussian processes (such as the tail approximations of the supremum) and extends the analysis to more complexed functionals.