Young Researchers Workshop: Ki-Net 2012-2019
The propagator norm of a (non symmetric, degenerate) linear Fokker-Planck equation
Vienna University of Technology
In this talk I will present the main theorem of my joined work with Anton
Arnold and Christian Schmeiser and its main two consequences. The result regards the L2 -propagator norm of a (possibly degenerate and non symmetric)
linear Fokker Planck equation in the context of hypocoercivity. This means that
the solution converges to the unique steady state with an exponential decay, but
paying a multiplicative constant greater than one.
It was actually the search of the optimal constant in these estimates that motivated the beginning of our investigation. The strategy consisted to find a link
between the large time behavior of FP equation to the behavior of its associated
ODE. The final result claims that the propagator norms of the two equations coincide.
This implies that all the properties of the convergence at the ODE level carry
over to the PDE level. For instance, the discussion of the optimality in the hypocoercive estimates for the PDE can be reduced to the study of the easier associated
ODE. The second consequences regards the so called hypocoercivity index. This
parameter describes the polynomial short time decay of the propagator norm of
a hypocoercive evolution. It is a immediate consequence of the main theorem
that this index is the same for both the PDE and the ODE, giving also a result of regularization in the sense of the Sobolev norm.