  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 $\dot x=-Cx$.
