Spectral analysis and long-time behaviour of a Fokker-Planck equation with a non-local perturbation

Abstract

In this article we consider a Fokker-Planck equation on ${\mathbb{R}}^d$ with a non-local, mass preserving perturbation. We first give a spectral analysis of the unperturbed Fokker-Planck operator in an exponentially weighted $L^2$-space. In this space the perturbed Fokker-Planck operator is an isospectral deformation of the Fokker-Planck operator, i.e. the spectrum of the Fokker-Planck operator is not changed by the perturbation. In particular, there still exists a unique (normalized) stationary solution of the perturbed evolution equation. Moreover, the perturbed Fokker-Planck operator generates a strongly continuous semigroup of bounded operators. Any solution of the perturbed equation converges 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 codimension $k$ (if $d=1$) in which the exponential decay rate of the semigroup equals $-k$.

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