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A. Arnold, Ch. Schmeiser, B. Signorello:
"Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift";
in: "ASC Report 5/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 37.

We are concerned with the short- and large-time behav-ior of the L2-propagator norm of Fokker-Planck equations with linear drift, i.e. ∂t f = divx (D∇x f +C x f ). With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as D = CS , the symmetric part of C . The main re-sult of this paper is the connection between normalized Fokker-Planck equations and their drift-ODE x˙ = −C x: Their L2-propagator norms actually coincide. This implies that optimal decay estimates on the drift-ODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential de-cay estimates of the Fokker-Planck solution towards the steady state. A second application of the theorem regards the short time behaviour of the solution: The short time regularization (in some weighted Sobolev space) is determined by its hypocoercivity index, which has recently been introduced for Fokker-Planck equations and ODEs (see [5, 1, 2]). In the proof we realize that the evolution in each invariant spectral subspace can be represented as an explicitly given, tensored version of the corresponding drift-ODE. In fact, the Fokker-Planck equation can even be considered as the second quantization of x˙ = −C x.

Fokker-Planck equation, large-time behavior, sharp exponential decay, semigroup norm, regularization rate, second quantization

http://www.asc.tuwien.ac.at/preprint/2020/asc05x2020.pdf