Abstract: We prove the monotonicity of the second-order moments of the discrete approximations arising from the Jordan-Kinderlehrer-Otto (JKO) variational scheme [2] for the heat equation as steepest descent with respect to a particular Monge-Kantorovich distance. This issue appears in the study of constrained optimization in the 2-Wasserstein metric performed by Carlen and Gangbo [1] via a duality argument. A direct method, using Lagrange multipliers, is outlined in [1] and provided here.

References

1. E. Carlen,W. Gangbo}, \emph{Constrained Steepest Descent in the 2-Wasserstein Metric}, Annal. Math., {\bf 157}, no. 3 (2003), 807-846.

2. R. Jordan, D. Kinderlehrer, F. Otto}, \emph{The Variational Formulation of the Fokker-Plank Equation}, SIAM J. of Math. Anal. {\bf 29} (1998), 1-17.