Rendiconti Lincei - Matematica e Applicazioni
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Published online: 2014-03-12
Spectral analysis and long-time behaviour of a Fokker-Planck equation with a non-local perturbationDominik Stürzer and Anton Arnold (1) Technische Universität Wien, Austria
(2) Technische Universität Wien, Austria
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 $-k$.
Keywords: Fokker-Planck, non-local perturbation, spectral analysis, exponential stability
Stürzer Dominik, Arnold Anton: 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), 53-89. doi: 10.4171/RLM/668