A gradient flow moves in the direction of steepest energy decrease. In Wasserstein space, the motion is a continuity equation whose velocity is the negative gradient of the first variation. The Jordan--Kinderlehrer--Otto scheme constructs this flow through repeated variational minimization.
Implicit Euler in a metric space
For a smooth function \(F\) on \(\mathbb R^d\), implicit Euler for \(\dot x=-\nabla F(x)\) is \[ x_{k+1}\in\operatorname*{arg\,min}_x \left\{F(x)+\frac{|x-x_k|^2}{2\tau}\right\}. \] The optimality condition \(\nabla F(x_{k+1})+(x_{k+1}-x_k)/\tau=0\) is the backward Euler equation. Replacing Euclidean distance by \(W_2\) gives a definition that requires no linear structure on measures.
The JKO minimizing movement
Definition. Given \(\mu_\tau^0\in\mathcal P_2\), define recursively \[ \mu_\tau^{k+1}\in \operatorname*{arg\,min}_{\mu\in\mathcal P_2} \left\{ \mathcal F(\mu)+\frac{1}{2\tau}W_2^2(\mu,\mu_\tau^k) \right\}. \] The piecewise-constant interpolation is \(\bar\mu_\tau(t)=\mu_\tau^k\) for \(t\in((k-1)\tau,k\tau]\).
Existence of one step. Suppose \(\mathcal F\) is proper, lower semicontinuous in \(W_2\), and its negative part is controlled so that the penalized functional is coercive in the second moment. Then every JKO step has a minimizer.
Proof. Take a minimizing sequence. The penalty and the triangle inequality against \(\delta_0\) bound its second moments, while the coercivity of \(\mathcal F\) prevents escape through its negative part. Wasserstein compactness yields a convergent subsequence. Lower semicontinuity of both \(\mathcal F\) and \(W_2^2(\cdot,\mu_\tau^k)\) makes the limit a minimizer.
Discrete energy estimates
Energy dissipation. Each JKO step satisfies \[ \mathcal F(\mu_\tau^{k+1}) +\frac{1}{2\tau}W_2^2(\mu_\tau^{k+1},\mu_\tau^k) \le\mathcal F(\mu_\tau^k). \] Hence, for \(N\tau\le T\), \[ \sum_{k=0}^{N-1}\frac{W_2^2(\mu_\tau^{k+1},\mu_\tau^k)}{\tau} \le2\{\mathcal F(\mu_\tau^0)-\inf\mathcal F\}. \]
Proof. Use \(\mu=\mu_\tau^k\) as a competitor in the minimization. Sum the resulting inequalities; the energy terms telescope.
Temporal modulus. For grid times \(s=m\tau<n\tau=t\), \[ W_2(\mu_\tau^m,\mu_\tau^n) \le\sum_{k=m}^{n-1}W_2(\mu_\tau^k,\mu_\tau^{k+1}) \le C\sqrt{t-s+\tau}. \] The second inequality follows from Cauchy--Schwarz and the discrete action bound.
Euler equation of one step
First-order condition. Suppose \(\mu_\tau^{k+1}=\rho\,\mathrm{d}x\), \(\mathcal F\) has first variation \(\delta\mathcal F/\delta\rho\), and \(T\) is the optimal map from \(\mu_\tau^{k+1}\) to \(\mu_\tau^k\). Then \[ \frac{T(x)-x}{\tau} = \nabla\!\left(\frac{\delta\mathcal F}{\delta\rho}(x)\right) \quad\text{for }\mu_\tau^{k+1}\text{-almost every }x. \]
Proof. Perturb the new measure by \(S_\varepsilon=\operatorname{id}+\varepsilon\xi\). The derivative of \(\mathcal F(S_{\varepsilon\#}\mu)\) is \[ \int\nabla\!\left(\frac{\delta\mathcal F}{\delta\rho}\right)\cdot\xi\,\mathrm{d}\mu. \] Coupling \(S_\varepsilon(x)\) with \(T(x)\) shows that the derivative of \(\tfrac12W_2^2(S_{\varepsilon\#}\mu,\mu_\tau^k)\) is \(\int(x-T(x))\cdot\xi(x)\,\mathrm{d}\mu\). Stationarity for every \(\xi\) gives the formula. Thus the forward discrete velocity \((x-T(x))/\tau\) is the negative Wasserstein gradient.
Metric slope and energy dissipation
Definition. The descending metric slope is \[ |\partial\mathcal F|(\mu) =\limsup_{\nu\to\mu} \frac{(\mathcal F(\mu)-\mathcal F(\nu))_+}{W_2(\mu,\nu)}. \] A curve of maximal slope satisfies the energy-dissipation inequality \[ \mathcal F(\mu_T) +\frac12\int_0^T|\dot\mu_t|^2\,\mathrm{d}t +\frac12\int_0^T|\partial\mathcal F|(\mu_t)^2\,\mathrm{d}t \le\mathcal F(\mu_0). \]
For sufficiently regular displacement-convex functionals, the inequality is an equality, and the minimal velocity satisfies \[ v_t=-\nabla\!\left(\frac{\delta\mathcal F}{\delta\rho_t}\right), \qquad \partial_t\rho_t =\nabla\cdot\left( \rho_t\nabla\frac{\delta\mathcal F}{\delta\rho_t} \right). \]
Passage to continuous time
Convergence of minimizing movements. Under properness, lower semicontinuity, compactness of energy sublevels, and an appropriate geodesic-convexity assumption, the interpolants \(\bar\mu_\tau\) converge as \(\tau\downarrow0\), locally uniformly in \(W_2\), to a curve of maximal slope for \(\mathcal F\).
Proof architecture. The discrete energy bound gives uniform second moments and a \(1/2\)-Hölder time modulus. A metric Arzelà--Ascoli theorem yields subsequential convergence. Lower semicontinuity passes the energy and discrete action to the limit. The variational inequality of each minimizing step controls the discrete slopes; their lower limit gives the metric-slope term. The limiting curve therefore satisfies the energy-dissipation inequality. Geodesic convexity then identifies it uniquely, so the whole family converges.
Evolution variational inequalities
Definition. An \(EVI_\lambda\) gradient flow satisfies, for every comparison measure \(\nu\), \[ \frac12\frac{\mathrm{d}}{\mathrm{d}t}W_2^2(\mu_t,\nu) +\frac{\lambda}{2}W_2^2(\mu_t,\nu) \le\mathcal F(\nu)-\mathcal F(\mu_t) \] for almost every \(t>0\).
Contraction and uniqueness. If two curves \(\mu_t,\nu_t\) solve the same \(EVI_\lambda\) flow, then \[ W_2(\mu_t,\nu_t)\le e^{-\lambda t}W_2(\mu_0,\nu_0). \] In particular, the flow is unique and depends continuously on its initial law.
Proof. Insert \(\nu_t\) as comparison in the EVI for \(\mu_t\), and \(\mu_t\) in the EVI for \(\nu_t\). The energy differences cancel. A standard doubling-of-time argument yields \[ \frac{\mathrm{d}}{\mathrm{d}t}W_2^2(\mu_t,\nu_t)+2\lambda W_2^2(\mu_t,\nu_t)\le0. \] Grönwall's inequality gives the result.