Heavy tailed large deviations for time averages of diffusions: the Ornstein-Uhlenbeck case

TL;DR

This paper investigates the large deviation behavior of time averages of the Ornstein-Uhlenbeck process raised to any power, revealing subexponential deviations and a non-convex rate function, with a new concise proof approach.

Contribution

It provides a self-contained, simplified proof of large deviations for powered Ornstein-Uhlenbeck processes, extending previous results with a novel approach.

Findings

Large deviations are subexponential beyond a critical power.

The rate function is non-convex and characterized by a Hamilton-Jacobi equation.

The proof is concise and relies on standard large deviations techniques.

Abstract

We study large deviations for the time average of the Ornstein-Uhlenbeck process raised to an arbitrary power. We prove that beyond a critical value, large deviations are subexponential in time, with a non-convex rate function whose main coefficient is given by the solution to a Hamilton-Jacobi problem. Although a similar problem was addressed in a recent work, the originality of the paper is to provide a short, self-contained proof of this result through a couple of standard large deviations arguments.