Wasserstein gradient flow for a feature-measure energy

Assume AC and the feature-measure energy hypotheses. Write $A{C}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2([0,\mathrm{\infty });{\mathcal{P}}_2({\mathbb{R}}^d))$ for curves whose restriction to every finite interval $[0,T]$ belongs to $A{C}^2([0,T];{\mathcal{P}}_2({\mathbb{R}}^d))$ in the $W_2$ metric. For $\mu \in {\mathcal{P}}_2(\mathrm{\Omega })$ with finite energy, define $F^{\mathrm{\prime }}(\mu )(u)=d{R}_{m_{\mu }}[\mathrm{\Phi }(u)]+V(u)$ on $\mathrm{\Omega }$, and extend this potential by $+\mathrm{\infty }$ off $\mathrm{\Omega }$. Its constrained subdifferential at $u$ is the local Fréchet subdifferential of $F^{\mathrm{\prime }}(\mu )+{\iota }_{\mathrm{\Omega }}$ defined in the feature-measure-energy convention. A Wasserstein gradient flow from ${\mu }_0\in {\mathcal{P}}_2(\mathrm{\Omega })$ is a curve $t\mapsto {\mu }_t$ in $A{C}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2([0,\mathrm{\infty });({\mathcal{P}}_2({\mathbb{R}}^d),W_2))$, with ${\mu }_{t=0}={\mu }_0$, for which there is a Borel velocity $v_t(u)$ satisfying the finite-action continuity equation and $v_t(u)$ equals the negative minimum-norm element of $\mathrm{\partial }(F^{\mathrm{\prime }}({\mu }_t)+{\iota }_{\mathrm{\Omega }})(u)$ for $dtd{\mu }_t(u)$-almost every $(t,u)$. This definition does not assert that the energy is displacement convex.

Feature-measure energy and its regularity assumptions

Absolutely continuous curves of order two in Wasserstein space

Finite-action continuity equation in Euclidean space

Chizat and Bach, Proposition 2.5