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

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.