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# 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

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n4.a5

#### Authors

#### Abstract

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 $\dot{x}=-Cx$.

#### Keywords

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