The Wick-Square Potential Relative to a Diagonal Gaussian Measure

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector and let $\beta=(\beta_{1},\dots,\beta_{d})\in\mathbb{R}^{d}$.

(The Wick-square potential)¶ The Wick-square potential with variances $c$ and couplings $\beta$ is the function

$$V_{c,\beta}:\mathbb{R}^{d}\to\mathbb{R},\qquad V_{c,\beta}(x)=\sum_{i=1}^{d}\beta_{i}\bigl(x_{i}^{2}-c_{i}\bigr).$$