Utilizing the fact that the marginal velocity can be written asu t(x) = R ut(x|z) pt(x|z)q(z) pt(x) dzwe get the required result: ∂pt(x) ∂t =−∇ ·(u t(x)pt(x))

for the conditional probability, we get ∂pt(x) ∂t = Z −∇ ·(u t(x|z)p t(x|z))q(z)dz = Z −∇ ·(u t(x|z)p t(x|z)q(z))dz =−∇ · Z ut(x|z) pt(x|z)q(z) pt(x) pt(x)dz =−∇ · Z ut(x|z) pt(x|z)p(z) pt(x) dz pt(x) =−∇ · Z ut(x|z) pt(x|z)q(z) pt(x) dz pt · 2022