On the diagonal s = t, the Lagrangian equation implies vt,t(x) =b t(x),(25) i.e., the parameterized velocity recovers the probability flow drift

TheEulerian equation:for all(s, t)∈[0,1] 2, for allx∈R d, ∂sXs,t(x) +∇X s,t(x)b s(x) = 0