Resonances for a diffusion with small noise

TL;DR

This paper investigates the spectral properties of a diffusion generator with small noise in , revealing the existence of resonances with real parts similar to fast-growing potential cases, despite the potential's slow growth.

Contribution

It extends the understanding of spectral resonances to cases where the potential grows slowly at infinity, contrasting with the well-studied fast growth scenarios.

Findings

Spectrum is R+ for slow growth potential.

Existence of resonances with real parts akin to fast growth case.

Imaginary parts of resonances are small.

Abstract

We study resonances for the generator of a diffusion with small noise in $R^d$ :$ L_\epsilon = -\epsilon\Delta + \nabla F \cdot \nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small.