Propagator norm and sharp decay estimates for FokkerPlanck equations with linear drift
Abstract
We are concerned with the short and largetime behavior of the $L^2$propagator norm of FokkerPlanck 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 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. 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 FokkerPlanck 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 driftODE. In fact, the FokkerPlanck equation can even be considered as the second quantization of $\dot x=Cx$.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.01405
 arXiv:
 arXiv:2003.01405
 Bibcode:
 2020arXiv200301405A
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 2 figures