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

TL;DR

This paper establishes a connection between Fokker-Planck equations with linear drift and their associated drift-ODEs, providing sharp decay estimates and insights into short-time regularization based on hypocoercivity.

Contribution

It proves that the $L^2$-propagator norm of normalized Fokker-Planck equations equals that of the drift-ODE, enabling transfer of decay estimates and analyzing short-time behavior.

Findings

$L^2$-propagator norms coincide for normalized equations and drift-ODEs

Decay estimates for drift-ODEs apply to Fokker-Planck solutions

Short-time regularization is characterized by hypocoercivity index

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. $\partial_t f=\mathrm{div}_{x}{(D \nabla_x f+Cxf)}$. 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 3.4) 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 [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 $\dot x=-Cx$.