Confining Potentials on Euclidean Space

In the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d$ be a natural number with $1\le d$. The set $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and convex, every convex combination of two of its points being again one of its points. For a function $V:\mathbb{R}^{d}\to\mathbb{R}$ of class $C^{2}$ on $\mathbb{R}^{d}$, $DV(x)$ is its gradient and $\Delta V(x)$ its Laplacian at $x$. Let $V:\mathbb{R}^{d}\to\mathbb{R}$.

(Confining potential)¶ The function $V$ is a confining potential if it is of class $C^{2}$ on $\mathbb{R}^{d}$ and convex on $\mathbb{R}^{d}$, and the following three conditions hold.

(a) (Superquadratic growth)¶ For every positive $M\in\mathbb{R}$ there is a positive $K\in\mathbb{R}$ with $M\lVert x\rVert^{2}\le V(x)$ for every $x\in\mathbb{R}^{d}$ with $K\le\lVert x\rVert$.

(b) (Slope bound)¶ There is $C\in\mathbb{R}$ with $\lVert DV(x)\rVert\le C\bigl(1+|V(x)|\bigr)$ for every $x\in\mathbb{R}^{d}$.

(c) (Curvature small against the potential)¶ For every positive $\varepsilon\in\mathbb{R}$ there is $C_{\varepsilon}\in\mathbb{R}$ with $\Delta V(x)\le\varepsilon|V(x)|+C_{\varepsilon}$ for every $x\in\mathbb{R}^{d}$.