A variational principle for the Kramers equation with unbounded external forces (Q1588434)

The Cauchy problem for the Kramers (or Kolmogorov-Fokker-Planck) equation with unbounded \(C^2\) external force fields is investigated via a time discrete variational principle. This principle arises from the idea of maximizing a relative entropy functional (relative to the stationary solution) with respect to the Kantorovich functional associated with a suitably chosen cost function. The scheme is shown to be convergent, and global existence of weak solutions for a large class of force fields and initial data follows. A steepest descent interpretation of the dynamics given by the Kramers dynamics with respect to the Kantorovich functional is offered.