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

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