Abstract: 
We are concerned with the short and largetime behavior of FokkerPlanck 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 FokkerPlanck equations and their driftODE $\dot x=Cx$: Their $L^2$propagator norms actually coincide. This implies that optimal decay estimates on the driftODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential decay estimates of the FokkerPlanck solution towards the steady state.
Secondly, we define an index of hypocoercivity, both for ODEs and FokkerPlanck 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. 
