Appl Math Optim 42:203–227 (2000)
2000 Springer-Verlag New York Inc.
Dynamical Systems in the Variational Formulation of the
Fokker–Planck Equation by the Wasserstein Metric
Department of Mathematics, Hokkaido University,
Sapporo 060-0810, Japan
Communicated by M. Nisio
Abstract. R. Jordan, D. Kinderlehrer, and F. Otto proposed the discrete-time ap-
proximation of the Fokker–Planck equation by the variational formulation. It is de-
termined by the Wasserstein metric, an energy functional, and the Gibbs–Boltzmann
entropy functional. In this paper we study the asymptotic behavior of the dynami-
cal systems which describe their approximation of the Fokker–Planck equation and
characterize the limit as a solution to a class of variational problems.
Key Words. Fokker–Planck equation, Wasserstein metric, Energy functional,
Gibbs–Boltzmann entropy functional, Dynamical systems, Variational problem.
AMS Classiﬁcation. Primary 60F15, Secondary 60H30.
We consider a nonnegative solution of the following Fokker–Planck equation:
∂p(t, x)/∂t =
p(t, x) + div
(∇(x)p(t, x)) (t > 0, x ∈ R
p(t, x) dx = 1 (t ≥ 0). (1.2)
Here (x) is a function from R
to R, and we put
(·) ≡∇, ·. In Nelson’s stochastic mechanics (see  and ), it is crucial to