Wasserstein gradient flows and JKO time steps

Let E be an energy on P₂, choose a positive time step τ, and suppose the displayed minimization has a solution with finite old energy. The next measure minimizes energy plus W₂ squared from the old measure divided by 2τ. The old measure itself is a competitor with zero transport penalty. Therefore the new energy plus its transport penalty is at most the old energy, yielding both an energy decrease and a quantitative bound on one-step movement.

This inequality does not guarantee minimizers exist for arbitrary energies. Properness, lower semicontinuity, and coercivity or compactness conditions enter the direct method, while suitable displacement convexity can supply uniqueness and stable limiting behavior. Nor does energy decrease alone prove that discrete trajectories converge to a particular PDE. One must control the accumulated transport movement and identify a limit through the energy’s variation and the conservation structure.

For an absolutely continuous probability density r, use the entropy convention ∫r log r, which is negative Shannon differential entropy. Add the confining potential energy ∫|x|²r/2. Assign infinite entropy to singular measures. Starting from a P₂ probability density with finite energy, the classical JKO setting for this smooth quadratic confinement gives convergence of suitable time interpolants as τ tends to zero to the Fokker–Planck evolution. This is a statement about the vanishing-step sequence, not exact equality after one finite minimization.

Formally the energy variation is log r + 1 + |x|²/2. Transport down its gradient gives velocity −∇log r − x; inserting this into mass conservation produces Δr + div(xr). The logarithmic term spreads mass while confinement pulls it inward. The formal differentiation explains the equation, while existence and convergence require the weak analytic theorem. Gaussian stationary density is consistent with this balance because its logarithmic gradient is −x.

For the different energy E(ρ) = one-half its second moment, atoms are allowed and there is no diffusion term. Couple an old position x with a candidate new position y. The one-step objective integrates y²/2 + |y − x|²/(2τ). Minimizing this strictly convex quadratic pointwise gives y = x/(1 + τ). The resulting deterministic pushforward achieves the lower bound for all couplings, proving the exact measure update.

Iterating this pure-potential step contracts each position by (1 + τ) to the negative k-th power. At fixed elapsed time kτ, letting τ shrink gives exponential contraction e to the minus time, matching the corresponding nondiffusive transport flow. A finite τ update remains implicit Euler, whereas explicit gradient descent uses factor 1 − τ and may overshoot. The entropy-free calculation cannot be used to claim that a Gaussian’s variance collapses under the entropy-plus-confinement Fokker–Planck equation.