  We derive a relation between the dissipation in a stochastic dynamics and the
Wasserstein distance. We show that the minimal amount of dissipation required
to transform an initial state to a final state during a diffusion process is
given by the Wasserstein distance between the two states, divided by the total
time of the process. This relation implies a lower bound on the dissipation for
any diffusion process in terms of its initial and final state. Using a lower
bound on the Wasserstein distance, we further show that we can give a lower
bound on the dissipation in terms of only the mean and convariance matrix of
the initial and final state. We apply this result to derive the optimal forces
that minimize the dissipation for given initial and final mean and covariance.
