= Confinement energy
{title2=$\mathcal V[\mu]=\int V(x)\,d\mu(x)$}
The average of a <confinement potential> against a <probability measure>. A potential bounded below with at most quadratic growth has finite energy on <probability measures> with finite second <absolute moments>. A lower <Hessian matrix> bound $D^2V\geq\lambda I$ gives $\lambda$-<displacement convexity>: apply <convexity> of $V-\lambda|\cdot|^2/2$ along each particle segment in an optimal <transport plan>, then integrate. Quadratic growth of $V$ alone does not justify two differentiations under the energy integral.
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