Spectral Analysis and Long-Time Behaviour of a Fokker-Planck Equation with a Non-Local Perturbation

TL;DR

This paper analyzes a Fokker-Planck equation with a non-local perturbation, showing that the spectral properties and exponential convergence to equilibrium are preserved, and characterizing the spectrum in weighted spaces.

Contribution

It demonstrates that the spectrum remains unchanged under a non-local perturbation and provides a detailed spectral characterization in weighted spaces.

Findings

The perturbed operator generates a $C_0$-semigroup.

The spectrum is unaffected by the perturbation.

Exponential convergence to the stationary state at rate -1.

Abstract

In this article we consider a Fokker-Planck equation with a non-local, mass preserving perturbation. We show that the perturbed Fokker-Planck operator generates a $C_0$-semigroup on an exponentially weighted $L^2$-space. Surprisingly, the spectrum of the Fokker-Planck operator is not affected by the perturbation. In particular there still exists a unique (normalized) stationary solution of the perturbed equation. And we have convergence towards the stationary state with exponential rate -1, the same rate as for the unperturbed Fokker-Planck equation. Moreover, for any $k\in\mathbb N$ there exists an invariant subspace with finite codimension in which the exponential decay rate equals $-k$. As a byproduct of our analysis we characterize the spectrum of the Fokker-Planck operator in $L^2$-spaces with exponential weights.