In this article we consider a Fokker-Planck equation on with a non-local, mass preserving perturbation. We first give a spectral analysis of the unperturbed Fokker-Planck operator in an exponentially weighted -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 , the same rate as for the unperturbed Fokker-Planck equation. Moreover, for any there exists an invariant subspace with codimension (if ) in which the exponential decay rate of the semigroup equals .
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Dominik Stürzer, Anton Arnold, Spectral analysis and long-time behaviour of a Fokker-Planck equation with a non-local perturbation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), no. 1, pp. 53–89DOI 10.4171/RLM/668