Non-spectral relaxation in one dimensional Ornstein-Uhlenbeck processes

TL;DR

This paper investigates how certain initial conditions in Ornstein-Uhlenbeck processes lead to non-spectral relaxation patterns, especially under Lévy noise, challenging the traditional spectral analysis approach.

Contribution

It demonstrates that non-spectral relaxation occurs for a broad class of initial conditions in generalized Ornstein-Uhlenbeck processes driven by Lévy noise, extending understanding beyond classical spectral theory.

Findings

Non-spectral relaxation occurs for initial conditions outside the domain of attraction of the stable distribution.

Spectral relaxation rates only apply when initial conditions are within the domain of attraction.

Generalized Lévy Ornstein-Uhlenbeck processes exhibit non-spectral relaxation as a common feature.

Abstract

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of an Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized L\'evy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to non-spectral relaxation can be considered exotic, for generalized OUPs driven by L\'evy noise these are the rule.