  We study a relationship between optimal transport theory and stochastic
thermodynamics for the Fokker-Planck equation. We show that the lower bound on
the entropy production is the action measured by the path length of the
$L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a
geometric measure of optimal transport theory, our result implies a geometric
interpretation of the entropy production. Based on this interpretation, we
obtain a thermodynamic trade-off relation between transition time and the
entropy production. This thermodynamic trade-off relation is regarded as a
thermodynamic speed limit which gives a tighter bound of the entropy
production. We also discuss stochastic thermodynamics for the subsystem and
derive a lower bound on the partial entropy production as a generalization of
the second law of information thermodynamics. Our formalism also provides a
geometric picture of the optimal protocol to minimize the entropy production.
We illustrate these results by the optimal stochastic heat engine and show a
geometrical bound of the efficiency.
