Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d$ be a natural number with $1\le d$, let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient $DV(x)$, partial derivatives $\partial_{i}V$ and $\partial_{j}\partial_{i}V$, Hessian matrix $D^{2}V(x)\in\mathcal{S}(d)$ and Laplacian $\Delta V$, and let $\nabla V:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the gradient map $x\mapsto DV(x)$. $\exp$ is the exponential function, $0_{d}$ the zero matrix of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, integrability is that of that definition, and $\mathcal{P}_{2}(\mathbb{R}^{d})$ is the set of probability measures with finite second moment.

1. (Regularity)¶ The functions $V$, $\Delta V$, $\partial_{i}V$ and $\partial_{j}\partial_{i}V$ ($i,j\in[d]$) are continuous on $\mathbb{R}^{d}$, hence Borel, and $\nabla V$ is Borel.

2. (Quadratic minorants)¶ For every positive $M\in\mathbb{R}$ there is $C_{M}\in\mathbb{R}$ with $M\lVert x\rVert^{2}-C_{M}\le V(x)$ for every $x\in\mathbb{R}^{d}$.

3. (Tangent inequality)¶ $V(x)+DV(x)\cdot(y-x)\le V(y)$ for all $x,y\in\mathbb{R}^{d}$.

4. (Hessian bounds)¶ For every $x\in\mathbb{R}^{d}$ and all $i,j\in[d]$,

5. (Growth under translation)¶ There are $v_{0}\in\mathbb{R}$ and a nonnegative $C\in\mathbb{R}$ such that $v_{0}\le V(x)$ and

6. (Integrability)¶ Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$ be such that $V$ is integrable with respect to $\mu$. Then $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$; the functions $\lVert\nabla V\rVert$, $\Delta V$ and $\partial_{j}\partial_{i}V$ ($i,j\in[d]$) are integrable with respect to $\mu$; and for every $a\in\mathbb{R}^{d}$ the function $x\mapsto V(x+a)$ is integrable with respect to $\mu$.