Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

Propagator norm and sharp decay estimates for Fokker–Planck equations with linear drift

Pages: 1047 – 1080



Anton Arnold (Institute for Analysis and Scientific Computing, Technische Universität Vienna, Austria)

Christian Schmeiser (Faculty of Mathematics, University of Vienna, Austria)

Beatrice Signorello (Institute for Analysis and Scientific Computing, Technische Universität Vienna, Austria)


We are concerned with the short- and large-time behavior of the $L^2$-propagator norm of Fokker–Planck equations with linear drift, i.e. $\partial_t f = \operatorname{div}_x (D \nabla_x f + Cxf)$. With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as $D=C_S$, the symmetric part of $C$. The main result of this paper (Theorem 3.1) is the connection between normalized Fokker–Planck equations and their drift-ODE $\dot{x}=-Cx$: Their $L^2$-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 decay 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 [F. Achleitner, A. Arnold, and E. Carlen, Kinet. Relat. Models 11, 4:953–1009, 2018]; [F. Achleitner, A. Arnold, and E. Carlen, arXiv preprint, arXiv:2109.10784, 2021]; [A. Arnold and J. Erb, arXiv preprint, arXiv:1409.5425v2, 2014]).

In the proof we realize that the evolution in each invariant spectral subspace can be representedas an explicitly given, tensored version of the corresponding drift-ODE. In fact, the Fokker–Planckequation can even be considered as the second quantization of $\dot{x}=-Cx$.


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

2010 Mathematics Subject Classification

35B40, 35H10, 35Q82, 35Q84, 47D07

Received 2 March 2020

Received revised 23 September 2021

Accepted 28 October 2021

Published 11 April 2022