|We are concerned with the short- and large-time behavior of Fokker-Planck equations with linear drift, i.e. $\partial_t f=div(D \nabla_x f+Cxf)$. A coordinate transformation can normalize these equations such that the diffusion and drift matrices are linked as $D=C_s$, the symmetric part of $C$.
The first main result of this talk 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.
Secondly, we define an index of hypocoercivity, both for ODEs and Fokker-Planck equations that describes the interplay between between the dissipative and conservative part of their generator. This index characterizes the polynomial decay of the propagator norm for short time.