Interaction potential ($W:{\mathbb{R}}^d\to \mathbb{R}$)

A function assigning energy to the separation of two particles. The common symmetric case is an even function, $W(z)=W(-z)$. Its average over pairs defines an interaction energy. If the Hessian matrix is bounded, the gradient is globally Lipschitz and grows at most linearly; its convolution with a probability measure therefore needs only a finite first absolute moment, even though the potential itself can require a second absolute moment.

For a twice continuously differentiable interaction potential with $\Vert D^{2}W\Vert \le L$ and a probability measure with finite first absolute moment, this subtraction produces a finite potential even if $W\ast \mu$ is infinite. Taylor's formula and linear growth of the gradient giveDominated differentiation under the integral sign gives $\nabla V_{\mathrm{ren}}=(\nabla W)\ast \mu$ and $D^2{V}_{\mathrm{ren}}=(D^{2}W)\ast \mu$. If the unrenormalized convolution is finite, the two potentials differ only by a constant, so they generate the same velocity field when that field depends on their gradient.

The factor $1/2$ counts unordered pairs once. For a symmetric interaction potential, this energy is unchanged by translating the entire probability measure. It therefore cannot have positive uniform displacement convexity on the unrestricted space containing translations. If $D^{2}W\ge \lambda I$, integrate the pointwise strong convexity inequality along an optimal transport plan and its independent copy. The variance identity for independent copies then gives $\lambda$-displacement convexity on every class with fixed barycenter of a measure, whenever the integrals are finite.