Real diffusion with complex spectral gap

TL;DR

This paper explores the spectral properties of Langevin process generators, demonstrating the existence of generators with non-real spectral gaps or Jordan block structures, extending understanding of their eigenvalue behavior.

Contribution

It introduces constructions of generators with complex spectral gaps or Jordan blocks, revealing new spectral phenomena beyond typical real eigenvalues.

Findings

Existence of generators with non-real spectral gaps

Construction of generators with Jordan block eigenvalues

Extension of spectral analysis in Langevin processes

Abstract

The low-lying eigenvalues of the generator of a Langevin process are known to satisfy the Eyring-Kramers law in the low temperature regime under suitable assumptions. These eigenvalues are generically real. We construct generators whose spectral gap is given by non-real eigenvalues or by a real eigenvalue having a Jordan block.