  In this paper, we construct the fundamental solution to a degenerate
diffusion of Kolmogorov type and develop a time-discrete variational scheme for
its adjoint equation. The so-called mean squared derivative cost function plays
a crucial role occurring in both the fundamental solution and the variational
scheme. The latter is implemented by minimizing a free energy functional with
respect to the Kantorovich optimal transport cost functional associated with
the mean squared derivative cost. We establish the convergence of the scheme to
the solution of the adjoint equation generalizing previously known results for
the Fokker-Planck equation and the Kramers equation.
