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

Each result cited below is universally quantified over the data in its own statement. By Confining Potentials on Euclidean Space §confining, $V$ is of class $C^{2}$ on the open convex set $\mathbb{R}^{d}$ and convex on it, and satisfies conditions (a), (b) and (c) there. By claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $V$ is also of class $C^{1}$ on $\mathbb{R}^{d}$.

Claim 1 (Regularity; claim 1). By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $V$ is continuous on $\mathbb{R}^{d}$. By clause 2 of C^k Maps on a Euclidean Open Set (with $k=1$), each $\partial_{i}V$ is of class $C^{1}$ on $\mathbb{R}^{d}$, hence continuous on $\mathbb{R}^{d}$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. By clause 1 of C^k Maps on a Euclidean Open Set, applied to the $C^{1}$ function $\partial_{i}V$, each $\partial_{j}\partial_{i}V$ (clause 4 there) exists at every point and is continuous at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces; since $d_{E}(x,y)$ is the nonnegative square root of $\sum_{k}(x_{k}-y_{k})^{2}$ by Euclidean Distance on $\mathbb{R}^n$, and the Euclidean distance of $\mathbb{R}^{1}$ is the absolute-value metric by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §distance, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field shows that this is the condition of Continuous Map Between Metric Spaces, so $\partial_{j}\partial_{i}V$ is continuous on $\mathbb{R}^{d}$. For the Laplacian $\Delta V=\sum_{i=1}^{d}\partial_{i}\partial_{i}V$ (The Laplacian of a Twice Continuously Differentiable Function §laplacian), let $(x_{k})_{k\in\mathbb{N}}$ converge to $x$ in $(\mathbb{R}^{d},d_{E})$; then $\partial_{i}\partial_{i}V(x_{k})\to\partial_{i}\partial_{i}V(x)$ for each $i$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, so $\Delta V(x_{k})\to\Delta V(x)$ by claim 1 of Arithmetic of Limits of Real Sequences, applied to the finitely many summands, and $\Delta V$ is continuous on $\mathbb{R}^{d}$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset. All these functions are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Finally $\nabla V(x)=DV(x)$ has components $\partial_{1}V(x),\dots,\partial_{d}V(x)$ by Gradient of a Real-Valued Function on a Euclidean Open Set, so $\nabla V$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps (the componentwise criterion, claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets).

Claim 2 (Quadratic minorants; claim 2). Let $M$ be positive, and let $K$ be positive with $M\lVert x\rVert^{2}\le V(x)$ whenever $K\le\lVert x\rVert$, by condition (a). The closed ball $\bar{B}=\bar{B}_{d_{E}}(0_{\mathbb{R}^{d}},K)=\{x:\lVert x\rVert\le K\}$ is nonempty by claim 1 of Elementary Properties of the Closed Ball in a Metric Space and compact by claim 2 of A Closed Euclidean Ball is Convex and Compact, and $V$ restricted to $\bar{B}$ has the continuity property required there by Claim 1; so Extreme Value Theorem on a Compact Subset of a Metric Space gives $x_{*}\in\bar{B}$ with $b=V(x_{*})\le V(x)$ for every $x\in\bar{B}$. Put $C_{M}=|MK^{2}-b|$, so that $0\le C_{M}$ and $MK^{2}-b\le C_{M}$ by claims 1 and 3 of Properties of the Absolute Value in an Ordered Field. If $\lVert x\rVert\le K$, then $\lVert x\rVert^{2}\le K^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $M\lVert x\rVert^{2}\le MK^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, and, since $-C_{M}\le-(MK^{2}-b)$ by sign reversal (claim 4 of Elementary Order Arithmetic in an Ordered Field) applied to $MK^{2}-b\le C_{M}$, adding these two inequalities (compatibility of the order with addition in Ordered Field, used twice) gives $M\lVert x\rVert^{2}-C_{M}\le MK^{2}-(MK^{2}-b)=b\le V(x)$. If $K\le\lVert x\rVert$, then $M\lVert x\rVert^{2}-C_{M}\le M\lVert x\rVert^{2}\le V(x)$. As the order of $\mathbb{R}$ is total, every $x$ falls under one of the two cases.

Claim 3 (Tangent inequality; claim 3). Let $x\in\mathbb{R}^{d}$. By A Real-Valued C^1 Function is Differentiable at Every Point, $V$ is differentiable at $x$ with derivative matrix the row matrix whose entry in column $i$ is $\partial_{i}V(x)$. By Elementary Calculus of the Subdifferential of a Convex Function §gradient, with $U=\mathbb{R}^{d}$ and $f=V$, the subdifferential of $V$ at $x$ relative to $\mathbb{R}^{d}$ is $\{g\}$ with $g_{i}=\partial_{i}V(x)$, that is $g=DV(x)$ by Gradient of a Real-Valued Function on a Euclidean Open Set. In particular $DV(x)$ is a subgradient, which by Subdifferential of a Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ §subdifferential means $V(x)+DV(x)\cdot(y-x)\le V(y)$ for every $y\in\mathbb{R}^{d}$.

Claim 4 (Hessian bounds; claim 4). Let $x\in\mathbb{R}^{d}$ and $A=D^{2}V(x)$, so $A_{ij}=\partial_{i}\partial_{j}V(x)$ by Hessian Matrix of a C^2 Function and $A_{ij}=A_{ji}$ by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian. The function $V$ satisfies the hypotheses of A Convex Function of Class $C^2$ has Positive Semidefinite Hessian with $U=\mathbb{R}^{d}$: the class $C^{2}$ used there asks, for a real-valued function, exactly continuity at every point of $V$, of each $\partial_{i}V$ and of each $\partial_{j}\partial_{i}V$, which is clauses 1 and 2 of C^k Maps on a Euclidean Open Set (the definition that supersedes it). Indeed, the older $C^{1}$ notion on which that $C^{2}$ notion is built asks, as clause 1 of C^k Maps on a Euclidean Open Set does with $m=1$, that the function be continuous at every point and that each of its partial derivatives exist at every point and be continuous at every point; and its partial derivatives are the same difference-quotient limits as in Partial Derivative on a Euclidean Open Set, whose extra requirement that the displaced point lie in the domain is automatic here since the domain is $\mathbb{R}^{d}$. Moreover the Hessian matrix used there has the same entries $\partial_{i}\partial_{j}V(x)$ as that of Hessian Matrix of a C^2 Function. Hence $0_{d}\preceq A$, that is, by The Positive Semidefinite Ordering on Symmetric Matrices, $0\le z\cdot(Az)$ for every $z\in\mathbb{R}^{d}$. With $z=e_{i}$ this gives $0\le A_{ii}$ for each $i$, so $\Delta V(x)=\sum_{i}A_{ii}$ (The Laplacian of a Twice Continuously Differentiable Function §laplacian) is nonnegative and $A_{ii}\le\Delta V(x)$, by claims 5 and 6 of Properties of Finite Sums. For $i\ne j$, moreover, $A_{ii}+A_{jj}=\sum_{l\in\{i,j\}}A_{ll}$ by claims 1 and 2 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set (with $F=\{i\}$ and the object $j\notin F$), and this is at most $\sum_{l\in[d]}A_{ll}=\Delta V(x)$ by claim 3 of Nonnegativity and Monotonicity of a Sum over a Finite Index Set and claim 1 of Properties of a Sum over a Finite Index Set; so $A_{ii}+A_{jj}\le\Delta V(x)$. For $i\ne j$ and $s\in\{1,-1\}$, $z=e_{i}+s\,e_{j}$ gives $0\le z\cdot(Az)=A_{ii}+2s\,A_{ij}+A_{jj}$, so $-(A_{ii}+A_{jj})\le2A_{ij}\le A_{ii}+A_{jj}$ and $|2A_{ij}|\le A_{ii}+A_{jj}\le\Delta V(x)$ by claim 6 of Properties of the Absolute Value in an Ordered Field; hence, since $|2A_{ij}|=|2|\,|A_{ij}|=2|A_{ij}|$ by claim 4 there (as $0<2$), multiplying by $\tfrac12=2^{-1}$, which is positive by claim 8 of Elementary Order Arithmetic in an Ordered Field, gives $|A_{ij}|\le\tfrac12\Delta V(x)$ (claim 5 of Elementary Arithmetic in an Ordered Field); and $\tfrac12\Delta V(x)\le\Delta V(x)$, by claim 8 of Elementary Order Arithmetic in an Ordered Field if $0<\Delta V(x)$ and trivially if $\Delta V(x)=0$. For $i=j$, $|A_{ii}|=A_{ii}\le\Delta V(x)$. Since $\partial_{j}\partial_{i}V(x)=A_{ji}=A_{ij}$, this proves $|\partial_{j}\partial_{i}V(x)|\le\Delta V(x)$ for all $i,j\in[d]$.

Claim 5 (Growth under translation; claim 5). Let $C_{1}$ be the constant of Claim 2 for $M=1$ and put $v_{0}=-C_{1}$; then $v_{0}\le\lVert x\rVert^{2}-C_{1}\le V(x)$ for every $x$. Put $W(y)=V(y)-v_{0}+1$, so $1\le W(y)$. Let $C_{b}$ be a constant as in condition (b). At $y=0_{\mathbb{R}^{d}}$ one has $0\le\lVert DV(y)\rVert\le C_{b}(1+|V(y)|)$ with $0<1+|V(y)|$, so $0\le C_{b}$ by claim 10 of Elementary Order Arithmetic in an Ordered Field. For every $y$, claims 1 and 5 of Properties of the Absolute Value in an Ordered Field give $|V(y)|=|W(y)+(v_{0}-1)|\le W(y)+|v_{0}-1|$, and since $1\le W(y)$, $1+|V(y)|\le(2+|v_{0}-1|)\,W(y)$; hence, by claim 5 of Elementary Arithmetic in an Ordered Field,

For $y\in\mathbb{R}^{d}$, $V$ is differentiable at $y$ with derivative matrix $B_{y}$, the row of the $\partial_{i}V(y)$, by A Real-Valued C^1 Function is Differentiable at Every Point; since $W(y+h)-W(y)-B_{y}h=V(y+h)-V(y)-B_{y}h$, $W$ is differentiable at $y$ with the same derivative matrix by Differentiability at a Point for Maps Between Euclidean Spaces, and $\partial_{i}W(y)=\partial_{i}V(y)$ by claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique. Fix $x,a\in\mathbb{R}^{d}$ and put $F(\tau)=W(x+\tau a)$ for $\tau$ in the open interval $(-1,2)$. By Chain Rule Along an Affine Path, $F$ is differentiable at every $\tau\in(-1,2)$ with $F'(\tau)=\sum_{i}\partial_{i}V(x+\tau a)\,a_{i}=DV(x+\tau a)\cdot a$, so by the Cauchy-Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product, claim 3 of Properties of the Absolute Value in an Ordered Field and the display, $F'(\tau)\le\lVert DV(x+\tau a)\rVert\,\lVert a\rVert\le C\lVert a\rVert\,F(\tau)$. Since $1\le F(\tau)$, $F(\tau)$ is an interior point of $(0,\infty)$, at which $\log$ is differentiable with derivative $F(\tau)^{-1}$ by The Natural Logarithm; by Chain Rule for One-Dimensional Derivatives, $G=\log\circ F$ is differentiable at every $\tau\in(-1,2)$ with $G'(\tau)=F'(\tau)\,F(\tau)^{-1}\le C\lVert a\rVert$, using claim 7 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field. By Mean Value Theorem on an Open Interval, applied on $(-1,2)$ to the points $0<1$, there is $\sigma\in(0,1)$ with $G(1)-G(0)=G'(\sigma)\le C\lVert a\rVert$. As $\exp$ is increasing with $\exp(u+v)=\exp(u)\exp(v)$ by claims 4 and 1 of Basic Properties of the Exponential Function, and $\exp(\log t)=t$ by The Natural Logarithm,

Claim 6 (Integrability; claim 6). Let $\mu$ be as there; then $\int|V|\,d\mu<\infty$ by Integrable Function and the Lebesgue Integral, and constant functions are integrable with respect to $\mu$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. In each case below a Borel function is dominated by a function of finite integral, so it is integrable by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and Integrable Function and the Lebesgue Integral, the dominating integrals being computed by claim 2 of Linearity and Monotonicity of the Lebesgue Integral and $\mu(\mathbb{R}^{d})=1$.

Second moment: with $C_{1}$ as in Claim 5, $\lVert x\rVert^{2}\le V(x)+C_{1}\le|V(x)|+|C_{1}|$ by Claim 2 and claim 3 of Properties of the Absolute Value in an Ordered Field, so $M_{2}(\mu)\le\int|V|\,d\mu+|C_{1}|<\infty$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment) and $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space.

Gradient: $\lVert\nabla V\rVert$ is Borel as the composite of $\nabla V$ (Claim 1) with the Borel norm (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and $\lVert\nabla V\rVert\le C_{b}(1+|V|)$ with $C_{b}\ge0$ as in Claim 5.

Laplacian and second derivatives: $\Delta V$ and $\partial_{j}\partial_{i}V$ are Borel by Claim 1; by Claim 4 and condition (c) with $\varepsilon=1$ there is $C'$ with $0\le\Delta V\le|V|+C'\le|V|+|C'|$, and $|\partial_{j}\partial_{i}V|\le\Delta V$.

Translates: let $a\in\mathbb{R}^{d}$ and $V_{a}(x)=V(x+a)$. If $x_{k}\to x$ in $(\mathbb{R}^{d},d_{E})$, then $x_{k}+a\to x+a$, as $d_{E}(x_{k}+a,x+a)=d_{E}(x_{k},x)$ by Euclidean Distance on $\mathbb{R}^n$, so $V_{a}(x_{k})\to V_{a}(x)$ by Claim 1 and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential; thus $V_{a}$ is continuous by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. With $E=\exp(C\lVert a\rVert)$, positive by claim 2 of Basic Properties of the Exponential Function, Claim 5 gives $v_{0}\le V_{a}(x)\le v_{0}-1+E\,(V(x)-v_{0}+1)$, and by claims 1, 3 and 5 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field, $-V_{a}(x)\le|v_{0}|$ and $V_{a}(x)\le(1+E)|v_{0}-1|+E\,|V(x)|$; so $|V_{a}(x)|\le|v_{0}|+(1+E)|v_{0}-1|+E\,|V(x)|$ by claim 6 there, a function of finite integral.