The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient $DV(y)$ and Laplacian $\Delta V(y)$ as in that definition, and let

be the $N$-particle potential of $V$. Then the following hold.

1. (Regularity)¶ $V_{N}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$, and for every $x\in\mathbb{R}^{dN}$

2. (Confinement)¶ $V_{N}$ is a confining potential on $\mathbb{R}^{dN}$.