Proof of Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement. The dimension $d$ is the one fixed in The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data; inside real expressions it is read in $\mathbb{R}$ through the canonical map $\iota$ of The Canonical Map from the Natural Numbers to a Field and written $d$ again, so that $0<d$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $\sqrt{d}$ is the nonnegative square root fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. Since $0\le M$, the real numbers $\sqrt{d}\,M$ and $dM$ are nonnegative by claim 5 of Elementary Arithmetic in an Ordered Field, and $(dM)^{2}$ is nonnegative by claim 2 of Nonnegativity of Squares in an Ordered Field. For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, as in The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings. Linearity, homogeneity and monotonicity of integrals are claim 1 (for measurable $[0,\infty]$-valued functions) and claim 2 (for integrable functions, together with $|\int g|\le\int|g|$) of Linearity and Monotonicity of the Lebesgue Integral; by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures a nonnegative Borel real function is integrated as a $[0,\infty]$-valued map, and every bounded Borel real function is integrable with respect to every probability measure on $\mathbb{R}^{q}$. A constant real function $c$ is Borel by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and its integral against a probability measure $\rho$ is $c$ when $0\le c$, by The Integral of an Indicator Function is the Measure of the Set (with the set $\mathbb{R}^{q}$, of measure $1$) and the homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral. We write $\varphi$ for $\varphi_{f}$.

Step 1. Regularity of $f$ and three pointwise bounds. The set $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Since $f$ is of class $C^{2}$, clause 2 of C^k Maps on a Euclidean Open Set (read through clause 3 there) shows that $f$ is of class $C^{1}$ and that each $\partial_{i}f$ is of class $C^{1}$ on $\mathbb{R}^{d}$, its partial derivatives being the iterated partial derivatives $\partial_{j}\partial_{i}f$ of clause 4 there; by clause 1 there the functions $f$, $\partial_{i}f$ and $\partial_{j}\partial_{i}f$ ($i,j\in[d]$) are continuous at every point of $\mathbb{R}^{d}$ in the Euclidean sense, hence continuous from $(\mathbb{R}^{d},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$ by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. For $x,y\in\mathbb{R}^{d}$ the segment joining $x$ and $y$ lies in $\mathbb{R}^{d}$, the Euclidean distance between $x$ and $y$ is $\lVert x-y\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $\lVert y-x\rVert=\lVert x-y\rVert$ by claim 5 there with the scalar $-1$, whose absolute value is $1$. We record three bounds, valid for all $x,y\in\mathbb{R}^{d}$ and all $i\in[d]$, and then the growth bounds they imply; no bound on the first derivatives of $f$ is used.

(T2) $|f(y)-f(x)-Df(x)\cdot(y-x)|\le C_{1}\lVert x-y\rVert^{2}$ with $C_{1}=2^{-1}dM$, by claim (ii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, read with $n=d$, $W=\mathbb{R}^{d}$ and $M_{2}=M$, since with $h=y-x$ the sum $\sum_{i=1}^{d}\partial_{i}f(x)h_{i}$ is $Df(x)\cdot(y-x)$ by Gradient of a Real-Valued Function on a Euclidean Open Set and Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$. Here $0\le C_{1}$, because $0<2^{-1}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field and $0\le dM$, using claim 5 of Elementary Arithmetic in an Ordered Field.

(T3) $|\partial_{i}f(y)-\partial_{i}f(x)|\le\sqrt{d}\,M\lVert x-y\rVert$, by claim (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder applied to the function $\partial_{i}f$, of class $C^{1}$ with partial derivatives $\partial_{j}\partial_{i}f$ bounded in absolute value by $M$.

(T4) $\lVert Df(x)-Df(y)\rVert^{2}\le(dM)^{2}\lVert x-y\rVert^{2}$. Indeed, by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, Gradient of a Real-Valued Function on a Euclidean Open Set and Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, $\lVert Df(x)-Df(y)\rVert^{2}=\sum_{i=1}^{d}(\partial_{i}f(x)-\partial_{i}f(y))^{2}$. Each summand is at most $(\sqrt{d}\,M\lVert x-y\rVert)^{2}=d\,M^{2}\lVert x-y\rVert^{2}$, by (T3), claim 2 of Properties of the Absolute Value in an Ordered Field, claim 1 of Nonnegativity of Squares in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Summing, by claims 2, 3 and 5 of Properties of Finite Sums applied to the nonnegative differences, and evaluating the sum of $d$ equal summands $d\,M^{2}\lVert x-y\rVert^{2}$ as $d\cdot d\,M^{2}\lVert x-y\rVert^{2}$ by claim 3 there and The Canonical Map from the Natural Numbers to a Field, we obtain (T4), the right-hand side being $(dM)^{2}\lVert x-y\rVert^{2}$ by commutativity.

(G1) $\lVert Df(x)-Df(0_{\mathbb{R}^{d}})\rVert\le dM\lVert x\rVert$ and $\lVert Df(x)\rVert\le\lVert Df(0_{\mathbb{R}^{d}})\rVert+dM\lVert x\rVert$ for every $x\in\mathbb{R}^{d}$. Indeed, $x-0_{\mathbb{R}^{d}}=x$ componentwise (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$), so (T4), read with $0_{\mathbb{R}^{d}}$ in place of $y$, gives $\lVert Df(x)-Df(0_{\mathbb{R}^{d}})\rVert^{2}\le(dM)^{2}\lVert x\rVert^{2}=(dM\lVert x\rVert)^{2}$. Both $\lVert Df(x)-Df(0_{\mathbb{R}^{d}})\rVert$ and $dM\lVert x\rVert$ are nonnegative (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and claim 5 of Elementary Arithmetic in an Ordered Field), so the first bound follows by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Since $Df(x)=(Df(x)-Df(0_{\mathbb{R}^{d}}))+Df(0_{\mathbb{R}^{d}})$ in the real vector space $\mathbb{R}^{d}$ (Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, claim 3 there), the triangle inequality Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle gives the second.

(G2) $|f(x)|\le|f(0_{\mathbb{R}^{d}})|+\lVert Df(0_{\mathbb{R}^{d}})\rVert\,\lVert x\rVert+C_{1}\lVert x\rVert^{2}$ for every $x\in\mathbb{R}^{d}$, where $C_{1}=\tfrac{dM}{2}$; with (G1) this is the pair of bounds asserted in claim 1. Indeed, (T2), read with $0_{\mathbb{R}^{d}}$ in place of $x$ and $x$ in place of $y$, gives $|r|\le C_{1}\lVert x\rVert^{2}$ for $r=f(x)-f(0_{\mathbb{R}^{d}})-Df(0_{\mathbb{R}^{d}})\cdot x$, as $x-0_{\mathbb{R}^{d}}=x$ and $\lVert0_{\mathbb{R}^{d}}-x\rVert=\lVert x\rVert$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §homogeneity with the scalar $-1$). As $f(x)=f(0_{\mathbb{R}^{d}})+Df(0_{\mathbb{R}^{d}})\cdot x+r$, two applications of Properties of the Absolute Value in an Ordered Field §triangle and the Cauchy-Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product, which gives $|Df(0_{\mathbb{R}^{d}})\cdot x|\le\lVert Df(0_{\mathbb{R}^{d}})\rVert\,\lVert x\rVert$, yield (G2).

(G3) $|\partial_{k}f(x)|\le\lVert Df(0_{\mathbb{R}^{d}})\rVert+\sqrt{d}\,M\lVert x\rVert$ for every $x\in\mathbb{R}^{d}$ and $k\in[d]$. Indeed, $\partial_{k}f(0_{\mathbb{R}^{d}})$ is the $k$th component of $Df(0_{\mathbb{R}^{d}})$ (Gradient of a Real-Valued Function on a Euclidean Open Set), so $|\partial_{k}f(0_{\mathbb{R}^{d}})|\le\lVert Df(0_{\mathbb{R}^{d}})\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate; and $|\partial_{k}f(x)-\partial_{k}f(0_{\mathbb{R}^{d}})|\le\sqrt{d}\,M\lVert x\rVert$ by (T3) read with $0_{\mathbb{R}^{d}}$ in place of $x$ and $x$ in place of $y$, as $\lVert0_{\mathbb{R}^{d}}-x\rVert=\lVert x\rVert$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §homogeneity with the scalar $-1$); Properties of the Absolute Value in an Ordered Field §triangle combines the two.

(G) With $A=|f(0_{\mathbb{R}^{d}})|+\lVert Df(0_{\mathbb{R}^{d}})\rVert+C_{1}$ and $B=\lVert Df(0_{\mathbb{R}^{d}})\rVert+\sqrt{d}\,M$, nonnegative real numbers (Properties of the Absolute Value in an Ordered Field §nonnegative, Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and claim 2 of Elementary Arithmetic in an Ordered Field), we have $|f(x)|\le A(1+\lVert x\rVert^{2})$ and $|\partial_{k}f(x)|\le B(1+\lVert x\rVert^{2})$ for every $x\in\mathbb{R}^{d}$ and $k\in[d]$. Indeed, we first show $1\le1+\lVert x\rVert^{2}$ and $\lVert x\rVert\le1+\lVert x\rVert^{2}$: as $0\le\lVert x\rVert^{2}$ (claim 2 of Nonnegativity of Squares in an Ordered Field), the first holds by claim 3 of Elementary Arithmetic in an Ordered Field; if $\lVert x\rVert\le1$ the second follows from the first; if $1<\lVert x\rVert$, then $0<\lVert x\rVert$ and $\lVert x\rVert=\lVert x\rVert\cdot1<\lVert x\rVert\cdot\lVert x\rVert$ by claims 2, 6 and 10 of Elementary Order Arithmetic in an Ordered Field, and $\lVert x\rVert^{2}\le1+\lVert x\rVert^{2}$ as $0\le1$. Multiplying the three inequalities $1\le1+\lVert x\rVert^{2}$, $\lVert x\rVert\le1+\lVert x\rVert^{2}$ and $\lVert x\rVert^{2}\le1+\lVert x\rVert^{2}$ by the nonnegative numbers $|f(0_{\mathbb{R}^{d}})|$, $\lVert Df(0_{\mathbb{R}^{d}})\rVert$ and $C_{1}$ respectively (claim 5 of Elementary Arithmetic in an Ordered Field) and adding them (claims 2 and 3 there) bounds the right-hand side of (G2) by $A(1+\lVert x\rVert^{2})$; multiplying the first two by $\lVert Df(0_{\mathbb{R}^{d}})\rVert$ and $\sqrt{d}\,M$ and adding bounds the right-hand side of (G3) by $B(1+\lVert x\rVert^{2})$. This proves (G).

Step 2. Claim 1. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, a probability measure on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. We first record a general fact.

(Q) Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $K$ be a nonnegative real number and let $h:\mathbb{R}^{d}\to\mathbb{R}$ be Borel with $|h(x)|\le K(1+\lVert x\rVert^{2})$ for every $x\in\mathbb{R}^{d}$. Then $h$ is integrable with respect to $\rho$. Indeed, the nonnegative Borel function $x\mapsto\lVert x\rVert^{2}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs) has $\rho$-integral $M_{2}(\rho)<\infty$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, so it is integrable by Integrable Function and the Lebesgue Integral; the constant $K$ is integrable, being bounded and Borel; hence $g_{0}(x)=K+K\lVert x\rVert^{2}$ is integrable by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and it is nonnegative. The function $|h|$ is Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and $|h|\le g_{0}$, so $\int|h|\,d\rho\le\int g_{0}\,d\rho<\infty$ by the monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and $h$ is integrable with respect to $\rho$ by Integrable Function and the Lebesgue Integral.

The function $f$ is Borel by Step 1, so (Q), read with $\rho=\mu$, $K=A$ and $h=f$, and (G) show that $f$ is integrable with respect to $\mu$. Thus $\varphi$ is defined. The gradient map $x\mapsto Df(x)$ has the components $\partial_{1}f,\dots,\partial_{d}f$ (Gradient of a Real-Valued Function on a Euclidean Open Set), each Borel by Step 1, so it is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; and $x\mapsto\lVert Df(x)\rVert^{2}$ is Borel as its composition with the Borel map $x\mapsto\lVert x\rVert^{2}$ (same clause and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs). For real $s,t$ we have $(s+t)^{2}\le2s^{2}+2t^{2}$, since $0\le(s-t)^{2}$; so (G1), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (applied to the nonnegative numbers $\lVert Df(x)\rVert$ and $\lVert Df(0_{\mathbb{R}^{d}})\rVert+dM\lVert x\rVert$) and claim 5 of Elementary Arithmetic in an Ordered Field give, for every $x\in\mathbb{R}^{d}$,

$$\lVert Df(x)\rVert^{2}\le2\lVert Df(0_{\mathbb{R}^{d}})\rVert^{2}+2(dM)^{2}\lVert x\rVert^{2}\le K_{D}\bigl(1+\lVert x\rVert^{2}\bigr),\qquad K_{D}=2\lVert Df(0_{\mathbb{R}^{d}})\rVert^{2}+2(dM)^{2}.$$

Here $K_{D}$ is a nonnegative real number: the squares $\lVert Df(0_{\mathbb{R}^{d}})\rVert^{2}$ and $(dM)^{2}$ are nonnegative by claim 2 of Nonnegativity of Squares in an Ordered Field, $2=1+1$ is nonnegative by claims 1 and 2 of Elementary Arithmetic in an Ordered Field, so $2\lVert Df(0_{\mathbb{R}^{d}})\rVert^{2}$ and $2(dM)^{2}$ are nonnegative by claim 5 there, and their sum $K_{D}$ is nonnegative by claim 2 there. The second inequality of the display holds because, as shown in (G), $1\le1+\lVert x\rVert^{2}$ and $\lVert x\rVert^{2}\le1+\lVert x\rVert^{2}$; multiplying these by the nonnegative numbers $2\lVert Df(0_{\mathbb{R}^{d}})\rVert^{2}$ and $2(dM)^{2}$ respectively (claim 5 of Elementary Arithmetic in an Ordered Field) and adding them (claims 2 and 3 there) bounds $2\lVert Df(0_{\mathbb{R}^{d}})\rVert^{2}+2(dM)^{2}\lVert x\rVert^{2}$ by $K_{D}(1+\lVert x\rVert^{2})$. By (Q), read with $\rho=\mu$, $K=K_{D}$ and $h=\lVert Df\rVert^{2}$, which is nonnegative and so equal to its absolute value, $\int_{\mathbb{R}^{d}}\lVert Df\rVert^{2}\,d\mu<\infty$. Hence the class of $Df$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, and it is again written $Df$. For $i,j\in[d]$ the entry of $D^{2}f(x)$ in row $i$ and column $j$ is $\partial_{i}\partial_{j}f(x)$ by Hessian Matrix of a C^2 Function; the function $\partial_{i}\partial_{j}f$ is Borel by Step 1 and bounded by $M$ by hypothesis, hence integrable with respect to $\mu$. Write $\Phi(\mu)=\int_{\mathbb{R}^{d}}D^{2}f\,d\mu$ for the matrix of these integrals. By claim 1 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian, read with $n=d$ and $U=\mathbb{R}^{d}$, $\partial_{i}\partial_{j}f(x)=\partial_{j}\partial_{i}f(x)$ for every $x$, so the entries of $\Phi(\mu)$ in positions $(i,j)$ and $(j,i)$ are integrals of the same function and coincide; hence $\Phi(\mu)\in\mathcal{S}(d)$ by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric. This proves claim 1.

Step 2a. Claim 3. Put $\lambda=dM$, a nonnegative real number with $\lambda\cdot2^{-1}=C_{1}$. The set $\mathbb{R}^{d}$ is convex, every point $t\,x+(1-t)\,y$ with $x,y\in\mathbb{R}^{d}$ lying in $\mathbb{R}^{d}$.

First, $f$ is semiconvex on $\mathbb{R}^{d}$ with constant $\lambda$. Let $x,y\in\mathbb{R}^{d}$ and $t\in\mathbb{R}$ with $0\le t\le1$, and put $z=t\,x+(1-t)\,y$ and $u=x-y$. Componentwise (Sum of Points of $\mathbb{R}^n$, Scalar Multiple of a Point of $\mathbb{R}^n$, Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$) we have $x-z=(1-t)\,u$, $y-z=(-t)\,u$, $z-x=(t-1)\,u$ and $z-y=t\,u$. Hence, with $p=Df(z)\cdot u$, the second argument rules of claim 5 of the bilinearity lemma give $Df(z)\cdot(x-z)=(1-t)\,p$ and $Df(z)\cdot(y-z)=-t\,p$, and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §homogeneity, with $|s|^{2}=s^{2}$ for real $s$ (Properties of the Absolute Value in an Ordered Field §nonnegative), gives $\lVert z-x\rVert^{2}=(1-t)^{2}\lVert u\rVert^{2}$ and $\lVert z-y\rVert^{2}=t^{2}\lVert u\rVert^{2}$. Now (T2), read with $z$ in place of $x$ and with $x$, respectively $y$, in place of $y$, together with Properties of the Absolute Value in an Ordered Field §two-sided, gives

$$f(x)\ge f(z)+(1-t)\,p-C_{1}(1-t)^{2}\lVert u\rVert^{2},\qquad f(y)\ge f(z)-t\,p-C_{1}t^{2}\lVert u\rVert^{2}.$$

Multiplying the first inequality by $t\ge0$ and the second by $1-t\ge0$ and adding, the terms in $p$ cancel and $t(1-t)^{2}+(1-t)t^{2}=t(1-t)$, so

$$f\bigl(t\,x+(1-t)\,y\bigr)\le t\,f(x)+(1-t)\,f(y)+\tfrac{\lambda}{2}\,t(1-t)\,\lVert x-y\rVert^{2}.$$

As $x$, $y$ and $t$ were arbitrary, Quadratic Increment Characterisation of Semiconvexity, read with $n=d$, $C=\mathbb{R}^{d}$ and the constant $\lambda$, shows that $f$ is semiconvex on $\mathbb{R}^{d}$ with constant $\lambda$. By Semiconvex Function on a Convex Subset of $\mathbb{R}^n$ (see also Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §convexity), the function $g:\mathbb{R}^{d}\to\mathbb{R}$, $g(x)=f(x)+\tfrac{\lambda}{2}\lVert x\rVert^{2}$, is convex on $\mathbb{R}^{d}$.

Second, $g$ is of class $C^{1}$ with $Dg(x)=Df(x)+\lambda x$. The matrix $\lambda I_{d}$ lies in $\mathcal{S}(d)$ and satisfies $z\cdot((\lambda I_{d})z)=\lambda\lVert z\rVert^{2}$ by Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix §identity, and $0_{\mathbb{R}^{d}}\cdot z=\sum_{k=1}^{d}0\cdot z_{k}=0$ by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ and claim 3 of Properties of Finite Sums. So Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, read with $n=d$, the matrix $\lambda I_{d}$ in place of the matrix written $M$ there, $q=0_{\mathbb{R}^{d}}$ and $c=0$, shows that $q_{\lambda}(z)=\tfrac{\lambda}{2}\lVert z\rVert^{2}$ defines a function of class $C^{2}$ on $\mathbb{R}^{d}$ with $Dq_{\lambda}(z)=(\lambda I_{d})z+0_{\mathbb{R}^{d}}=(\lambda I_{d})z$ (Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, claim 1 there). By Matrix-Vector Product, Scalar Multiple of a Real Matrix and Identity Matrix, the $i$th component of $(\lambda I_{d})z$ is a sum over $k\in[d]$ whose summands vanish for $k\ne i$ and whose $i$th summand is $\lambda z_{i}$, so it equals $\lambda z_{i}$ by claim 7 of Properties of Finite Sums; thus $\partial_{i}q_{\lambda}(z)=\lambda z_{i}$ by Gradient of a Real-Valued Function on a Euclidean Open Set. By clause 2 of C^k Maps on a Euclidean Open Set (read through clause 3), $q_{\lambda}$ and $f$ are of class $C^{1}$, so $g=f+q_{\lambda}$ is of class $C^{1}$ on $\mathbb{R}^{d}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, with $\partial_{i}g(x)=\partial_{i}f(x)+\lambda x_{i}$ by claim 1 there; by Gradient of a Real-Valued Function on a Euclidean Open Set, Sum of Points of $\mathbb{R}^n$ and Scalar Multiple of a Point of $\mathbb{R}^n$, $Dg(x)=Df(x)+\lambda x$ for every $x$.

Third, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu the elements of $L^{2}(\mu;\mathbb{R}^{d})$ are the classes of the square-integrable random vectors on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ in the sense of The Space of Square-Integrable Random Vectors §space. The gradient map $Df$ is such a random vector by Step 2, and so is $\mathrm{id}$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity; as these are closed under sums and scalar multiples (The Space of Square-Integrable Random Vectors §space), the gradient map $\nabla g:x\mapsto Dg(x)=Df(x)+\lambda x$ satisfies $\int_{\mathbb{R}^{d}}\lVert\nabla g\rVert^{2}\,d\mu<\infty$, and by The Space of Square-Integrable Random Vectors §classes its class is $\nabla g=Df+\lambda\,\mathrm{id}$ in $L^{2}(\mu;\mathbb{R}^{d})$. Since $g$ is of class $C^{1}$ and convex on $\mathbb{R}^{d}$, The Gradient of a Convex Continuously Differentiable Function Belongs to the Tangent Space When It Is Square-Integrable §tangent gives $\nabla g\in T_{\mu}$, and $\mathrm{id}\in T_{\mu}$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity. By Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed, $T_{\mu}$ is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$, so it contains $\nabla g+(-\lambda)\,\mathrm{id}=Df$. As $\mu$ was arbitrary, this proves claim 3.

Step 3. Integrals along a coupling. Let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, $(\mathrm{pr}_{1})_{\#}\pi=\mu$ and $(\mathrm{pr}_{2})_{\#}\pi=\nu$, the projections being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs. Hence, by Step 2 and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, the Borel functions $z\mapsto f(x)$ and $z\mapsto f(y)$ (compositions of Borel maps, Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps) are $\pi$-integrable with

$$\int_{\mathbb{R}^{d+d}}f(x)\,\pi(dz)=\varphi(\mu),\qquad\int_{\mathbb{R}^{d+d}}f(y)\,\pi(dz)=\varphi(\nu).$$

Moreover $I(\pi)=\int_{\mathbb{R}^{d+d}}\lVert x-y\rVert^{2}\,\pi(dz)$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost, a nonnegative real number by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite.

Step 4. Property (a). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(\mathbb{R}^{d})$ converging to $\mu$ in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions. By Convergent Sequence in a Metric Space, for every positive $\varepsilon$ there is $N$ with $W_{2}(\mu_{n},\mu)<\varepsilon$ for $n\ge N$; as $W_{2}(\mu_{n},\mu)$ is nonnegative (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric), it equals $|W_{2}(\mu_{n},\mu)-0|$ by Absolute Value in an Ordered Field, so $\lim_{n\to\infty}W_{2}(\mu_{n},\mu)=0$ in the sense of Limit of a Sequence of Real Numbers. The function $f$ is continuous (Step 1) and satisfies (G), so Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §quadratic, read with $m=d$ and $h=f$, gives $\lim_{n\to\infty}\varphi(\mu_{n})=\varphi(\mu)$; since $d_{\mathbb{R}}(s,t)=|s-t|$ (The Absolute Value Metric on the Real Line), this is convergence of $(\varphi(\mu_{n}))_{n}$ to $\varphi(\mu)$ in $(\mathbb{R},d_{\mathbb{R}})$ in the sense of Convergent Sequence in a Metric Space. As $\mu$ and the sequence were arbitrary, Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset, read with $X=A=\mathcal{P}_{2}(\mathbb{R}^{d})$ and the metric $W_{2}$ (whose restriction to $A$ is $W_{2}$ itself) and with $Y=\mathbb{R}$, shows that $\varphi$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$.

Step 5. Property (b) and the gradient. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Step 2, the gradient map $Df$ is Borel with $\int\lVert Df\rVert^{2}\,d\mu<\infty$ and its class $Df$ lies in $L^{2}(\mu;\mathbb{R}^{d})$; by Step 2a, $Df\in T_{\mu}$. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing, applied with the representative $Df$, the function $z\mapsto Df(x)\cdot(y-x)$ is Borel and $\pi$-integrable with integral $\mathcal{J}(Df,\pi)$. With Step 3 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the function

$$R_{\pi}(z)=f(y)-f(x)-Df(x)\cdot(y-x)$$

is $\pi$-integrable with $\int R_{\pi}\,d\pi=\varphi(\nu)-\varphi(\mu)-\mathcal{J}(Df,\pi)$, and $|\int R_{\pi}\,d\pi|\le\int|R_{\pi}|\,d\pi$. The function $|R_{\pi}|$ is nonnegative and Borel (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and $|R_{\pi}(z)|\le C_{1}\lVert x-y\rVert^{2}$ for every $z$ by (T2); so the monotonicity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, with Step 3, give

$$\bigl|\varphi(\nu)-\varphi(\mu)-\mathcal{J}(Df,\pi)\bigr|\le C_{1}\,I(\pi)\qquad\text{for all }\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})\text{ and }\pi\in\Pi(\mu,\nu). \tag{B1}$$

Now let $\varepsilon\in\mathbb{R}$ be positive. First put $\theta=\varepsilon\,(C_{1}+1)^{-1}$; it is positive, because $0<C_{1}+1$ by claims 3 and 6 of Elementary Order Arithmetic in an Ordered Field (as $0\le C_{1}$), and then claims 7 and 5 there apply. Then let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ satisfy $I(\pi)<\theta^{2}$, and put $s=\sqrt{I(\pi)}$, so that $0\le s$ and $s^{2}=I(\pi)$. By claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $s<\theta$. By claim 5 of Elementary Arithmetic in an Ordered Field, applied first to $s\le\theta$ with the nonnegative factor $C_{1}s$, and then to $C_{1}\le C_{1}+1$ with the nonnegative factor $\theta s$,

$$C_{1}\,I(\pi)=C_{1}\,s\,s\le C_{1}\,\theta\,s\le(C_{1}+1)\,\theta\,s=\varepsilon\,s .$$

With (B1), $|\varphi(\nu)-\varphi(\mu)-\mathcal{J}(Df,\pi)|\le\varepsilon\sqrt{I(\pi)}$. This is the condition of Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable with $\eta=Df$, so $\varphi$ is differentiable along couplings at $\mu$, and by Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §gradient its gradient along couplings is $\nabla\varphi(\mu)=Df$, which lies in $T_{\mu}$. As $\mu$ was arbitrary, property (b) holds with $Q=\mathcal{P}_{2}(\mathbb{R}^{d})$, and the gradient is as asserted in claim 2.

Step 6. Property (c). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\pi_{n}\in\Pi(\mu_{n},\mu)$ with $\lim_{n\to\infty}I(\pi_{n})=0$. By Step 5, $\nabla\varphi(\mu_{n})$ and $\nabla\varphi(\mu)$ are the classes of $Df$ in $L^{2}(\mu_{n};\mathbb{R}^{d})$ and in $L^{2}(\mu;\mathbb{R}^{d})$, so by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined, computed with the representative $Df$ on both sides, their discrepancy along $\pi_{n}$ is the nonnegative real number

$$\delta_{n}=\int_{\mathbb{R}^{d+d}}\lVert Df(x)-Df(y)\rVert^{2}\,\pi_{n}(dz).$$

By (T4), the monotonicity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and Step 3 (read with $\mu_{n}$ and $\mu$ in place of $\mu$ and $\nu$), $\delta_{n}\le(dM)^{2}I(\pi_{n})$. Let $\varepsilon\in\mathbb{R}$ be positive and put $C_{2}=(dM)^{2}+1$, positive as in Step 5. By Limit of a Sequence of Real Numbers choose $N$ with $|I(\pi_{n})-0|<\varepsilon\,C_{2}^{-1}$ for $n\ge N$; as $I(\pi_{n})\ge0$ this says $I(\pi_{n})<\varepsilon\,C_{2}^{-1}$. For $n\ge N$, by claim 5 of Elementary Arithmetic in an Ordered Field and claims 10 and 2 of Elementary Order Arithmetic in an Ordered Field,

$$|\delta_{n}-0|=\delta_{n}\le(dM)^{2}I(\pi_{n})\le C_{2}\,I(\pi_{n})<C_{2}\,\varepsilon\,C_{2}^{-1}=\varepsilon .$$

So $(\delta_{n})_{n}$ converges to $0$, which is property (c).

Step 7. The translated functional. For $b\in\mathbb{R}^{d}$ the translation $\tau_{b}(x)=x+b$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants is Borel: its $l$th component $x\mapsto x_{l}+b_{l}$ is the sum of a coordinate projection, Borel by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and a constant, so it is Borel by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and 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 applies. Consequently $g\circ\tau_{b}$ is Borel for every Borel $g:\mathbb{R}^{d}\to\mathbb{R}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$. Then $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields §translation, so $f$ is integrable with respect to it by Step 2, and by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward the function $f\circ\tau_{a}$ is $\mu$-integrable and

$$\phi_{\mu}(a)=\varphi\bigl((\tau_{a})_{\#}\mu\bigr)=\int_{\mathbb{R}^{d}}f(x+a)\,\mu(dx). \tag{P}$$

Step 8. Partial derivatives of averaged translates. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $g:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$, let $K$ and $L$ be nonnegative real numbers with $|\partial_{k}g(x)|\le K+L\lVert x\rVert$ for all $x\in\mathbb{R}^{d}$ and $k\in[d]$, and suppose that $g\circ\tau_{b}$ is $\mu$-integrable for every $b\in\mathbb{R}^{d}$. Put $\psi(a)=\int_{\mathbb{R}^{d}}g(x+a)\,\mu(dx)$. We show: for every $a\in\mathbb{R}^{d}$ and $i\in[d]$, the function $x\mapsto\partial_{i}g(x+a)$ is $\mu$-integrable, and $\partial_{i}\psi(a)$ exists and equals $\int_{\mathbb{R}^{d}}\partial_{i}g(x+a)\,\mu(dx)$. Fix $a$ and $i$, let $e_{i}$ be the standard basis vector of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis, and let $J=(-1,1)$, an open interval all of whose points are interior points by An Open Interval is an Interval All of Whose Points Are Interior. Apply Differentiation under the Integral Sign with the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$, the interval $J$ and $F(t,x)=g\bigl((x+a)+te_{i}\bigr)$, where $(x+a)+te_{i}=x+(a+te_{i})$ by the componentwise definitions Sum of Points of $\mathbb{R}^n$ and Scalar Multiple of a Point of $\mathbb{R}^n$. Condition (i) holds by hypothesis with $b=a+te_{i}$. For condition (ii), fix $x$: by claim 2 of Derivatives Along a Segment for C^1 Functions on a Euclidean Open Set, read with $n=d$, $W=\mathbb{R}^{d}$, the point $x+a$, the vector $h=e_{i}$ and the interval $J$ (the point written $x+\tau h$ there being the vector $(x+a)+\tau e_{i}$, by the same componentwise definitions), the function $t\mapsto F(t,x)$ is differentiable at every point $t$ of $J$ with derivative $\sum_{k=1}^{d}\partial_{k}g(x+a+te_{i})\,(e_{i})_{k}$, which equals $\partial_{i}g(x+a+te_{i})$ by claim 7 of Properties of Finite Sums, since $(e_{i})_{k}=0$ for $k\ne i$ and $(e_{i})_{i}=1$. So $D_{1}F(t,x)=\partial_{i}g(x+a+te_{i})$. For condition (iii), let $t\in J$, so that $-1<t<1$ and hence $|t|<1$ by Properties of the Absolute Value in an Ordered Field §strict-two-sided. By Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle (twice), Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §homogeneity and $\lVert e_{i}\rVert=1$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis), $\lVert x+a+te_{i}\rVert\le\lVert x\rVert+\lVert a\rVert+|t|\le\lVert x\rVert+\lVert a\rVert+1$. So, with $K_{a}=K+L(\lVert a\rVert+2)$, a nonnegative real number, and using $\lVert x\rVert\le1+\lVert x\rVert^{2}$ and $1\le1+\lVert x\rVert^{2}$ (Step 1, proof of (G)) together with claims 2, 3 and 5 of Elementary Arithmetic in an Ordered Field,

$$|D_{1}F(t,x)|\le K+L\bigl(\lVert x\rVert+\lVert a\rVert+1\bigr)\le K_{a}\bigl(1+\lVert x\rVert^{2}\bigr)\qquad(t\in J,\ x\in\mathbb{R}^{d}).$$

The function $x\mapsto K_{a}(1+\lVert x\rVert^{2})$ is nonnegative and Borel (claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs), and it is $\mu$-integrable by (Q) of Step 2, read with $\rho=\mu$ and $K=K_{a}$. The theorem gives that $x\mapsto\partial_{i}g(x+a+te_{i})$ is integrable for every $t\in J$, in particular (with $t=0$) that $x\mapsto\partial_{i}g(x+a)$ is, and that $t\mapsto\psi(a+te_{i})$ is differentiable at $0$ with derivative $\Lambda=\int\partial_{i}g(x+a)\,\mu(dx)$. By Derivative at an Interior Point, for every positive $\varepsilon$ there is a positive $\delta$ such that every real $h$ with $0<|h|<\delta$ and $h\in J$ satisfies $\bigl|h^{-1}(\psi(a+he_{i})-\psi(a))-\Lambda\bigr|<\varepsilon$. Replacing $\delta$ by the lesser of $\delta$ and $1$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), the requirement $h\in J$ follows from $|h|<1$ by claim 9 of Properties of the Absolute Value in an Ordered Field. The point $a+he_{i}$ has $i$th component $a_{i}+h$ and $k$th component $a_{k}$ for $k\ne i$ (Sum of Points of $\mathbb{R}^n$, Scalar Multiple of a Point of $\mathbb{R}^n$ and Euclidean Points as Tuples of Real Numbers), so this is exactly the condition of Partial Derivative on a Euclidean Open Set, with $U=\mathbb{R}^{d}$, that $\partial_{i}\psi(a)$ exists with value $\Lambda$.

Step 9. Continuity of averaged translates. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $g:\mathbb{R}^{d}\to\mathbb{R}$ be continuous at every point in the Euclidean sense, and let $K$ be a nonnegative real number with $|g(x)|\le K(1+\lVert x\rVert^{2})$ for every $x$. Then $g$ is Borel (as in Step 1). For $a\in\mathbb{R}^{d}$ put $\rho_{a}=(\tau_{a})_{\#}\mu$, which lies in $\mathcal{P}_{2}(\mathbb{R}^{d})$ (Step 7); by (Q) of Step 2, read with $\rho=\rho_{a}$ and $h=g$, the function $g$ is $\rho_{a}$-integrable, so by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward the Borel function $g\circ\tau_{a}$ (Step 7) is $\mu$-integrable and

$$\psi(a)=\int_{\mathbb{R}^{d}}g(x+a)\,\mu(dx)=\int_{\mathbb{R}^{d}}g\,d\rho_{a}$$

is defined for every $a$. We show that $\psi$ is continuous at every $b\in\mathbb{R}^{d}$ in the Euclidean sense. First, $W_{2}(\rho_{a},\rho_{b})=\lVert a-b\rVert$ for all $a,b\in\mathbb{R}^{d}$: by The Squared Wasserstein Distance to a Fixed Measure: a One-Sided Bound Along Couplings, Differentiability at a Uniquely Mapped Source, and the Translation and Centring Identities §translation, read with $\nu=\mu$,

$$W_{2}(\rho_{a},\rho_{b})^{2}=W_{2}(\mu,\mu)^{2}+2\,(a-b)\cdot\bigl(m(\mu)-m(\mu)\bigr)+\lVert a-b\rVert^{2}=\lVert a-b\rVert^{2},$$

because $W_{2}(\mu,\mu)=0$ by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation and $(a-b)\cdot(m(\mu)-m(\mu))=(a-b)\cdot m(\mu)-(a-b)\cdot m(\mu)=0$ by claim 5 of the bilinearity lemma; as $W_{2}(\rho_{a},\rho_{b})$ and $\lVert a-b\rVert$ are nonnegative (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square), claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the assertion. By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion, read with $X=A=\mathbb{R}^{d}$, the metric $d_{E}$ and $Y=\mathbb{R}$ with $d_{\mathbb{R}}$, it suffices to show that $\psi(a_{m})\to\psi(b)$ whenever $(a_{m})_{m\in\mathbb{N}}$ converges to $b$ in $(\mathbb{R}^{d},d_{E})$. Given such a sequence and a positive $\varepsilon$, Convergent Sequence in a Metric Space provides $N$ with $d_{E}(a_{m},b)<\varepsilon$ for $m\ge N$; since $d_{E}(a_{m},b)=\lVert a_{m}-b\rVert=W_{2}(\rho_{a_{m}},\rho_{b})$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance) and this number is nonnegative, it equals $|W_{2}(\rho_{a_{m}},\rho_{b})-0|$ by Absolute Value in an Ordered Field, so $\lim_{m\to\infty}W_{2}(\rho_{a_{m}},\rho_{b})=0$ in the sense of Limit of a Sequence of Real Numbers. The function $g$ is continuous for $d_{E}$ and $d_{\mathbb{R}}$ by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, so Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §quadratic, read with $m=d$, $h=g$, $A=K$, the measure $\rho_{b}$ and the sequence $(\rho_{a_{m}})_{m}$, gives $\lim_{m\to\infty}\psi(a_{m})=\psi(b)$; by The Absolute Value Metric on the Real Line this is convergence in $(\mathbb{R},d_{\mathbb{R}})$, as required.

Step 10. Property (d) and the Hessian at the origin. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Step 7, $f\circ\tau_{b}$ is $\mu$-integrable for every $b$, and $\phi_{\mu}$ is given by (P). Step 8, read with $g=f$, $K=\lVert Df(0_{\mathbb{R}^{d}})\rVert$ and $L=\sqrt{d}\,M$ (the bound on $\partial_{k}f$ being (G3)), shows that for all $a\in\mathbb{R}^{d}$ and $i\in[d]$ the partial derivative $\partial_{i}\phi_{\mu}(a)$ exists and equals $\psi_{i}(a)=\int\partial_{i}f(x+a)\,\mu(dx)$. Fix $i$. The function $\partial_{i}f$ is of class $C^{1}$ with partial derivatives $\partial_{j}\partial_{i}f$ bounded by $M$ (Step 1), and it is continuous at every point (Step 1) with $|\partial_{i}f(x)|\le B(1+\lVert x\rVert^{2})$ by (G), so each $(\partial_{i}f)\circ\tau_{b}$ is $\mu$-integrable by Step 9, read with $g=\partial_{i}f$ and $K=B$; so Step 8, read with $g=\partial_{i}f$, $K=M$ and $L=0$, shows that for all $a$ and $j\in[d]$ the partial derivative $\partial_{j}\psi_{i}(a)$ exists and equals $\psi_{ij}(a)=\int\partial_{j}\partial_{i}f(x+a)\,\mu(dx)$. Step 9, read with $g=f$ and $K=A$ (by (G); here $\psi=\phi_{\mu}$ by (P)), with $g=\partial_{i}f$ and $K=B$ (by (G)), and with $g=\partial_{j}\partial_{i}f$ and $K=M$ (continuous at every point by Step 1, and $|\partial_{j}\partial_{i}f(x)|\le M\le M(1+\lVert x\rVert^{2})$ by the hypothesis, $1\le1+\lVert x\rVert^{2}$ and claim 5 of Elementary Arithmetic in an Ordered Field), shows that $\phi_{\mu}$, $\psi_{i}$ and $\psi_{ij}$ are continuous at every point. By clause 1 of C^k Maps on a Euclidean Open Set (read through clause 3), $\phi_{\mu}$ is of class $C^{1}$, its partial derivatives $\partial_{i}\phi_{\mu}=\psi_{i}$ existing everywhere and being continuous, and each $\psi_{i}$ is of class $C^{1}$, its partial derivatives $\psi_{ij}$ existing everywhere and being continuous; by clause 2 there with $k=1$, $\phi_{\mu}$ is of class $C^{2}$ on $\mathbb{R}^{d}$. This is property (d). In the notation of clause 4 there, $\partial_{j}\partial_{i}\phi_{\mu}=\psi_{ij}$; so by Hessian Matrix of a C^2 Function the entry of $D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})$ in row $i$ and column $j$ is $\partial_{i}\partial_{j}\phi_{\mu}(0_{\mathbb{R}^{d}})=\psi_{ji}(0_{\mathbb{R}^{d}})=\int\partial_{i}\partial_{j}f\,d\mu$, as $x+0_{\mathbb{R}^{d}}=x$ (Sum of Points of $\mathbb{R}^n$). This is the corresponding entry of $\Phi(\mu)$, so, matrices with the same entries being equal (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices),

$$D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})=\Phi(\mu)=\int_{\mathbb{R}^{d}}D^{2}f\,d\mu\qquad\text{for every }\mu\in\mathcal{P}_{2}(\mathbb{R}^{d}). \tag{H}$$

Step 11. Finite sums of null sequences. Let $(c^{k}_{n})_{n\in\mathbb{N}}$, for $k\in[d]$, be sequences of real numbers each converging to $0$. We show that $\bigl(\sum_{k=1}^{d}c^{k}_{n}\bigr)_{n\in\mathbb{N}}$ converges to $0$. Let $\mathcal{A}$ be the set of natural numbers $m$ such that either $m\notin[d]$ or the sequence $\bigl(\sum_{k=1}^{m}c^{k}_{n}\bigr)_{n}$ converges to $0$. Then $1\in\mathcal{A}$, since $\sum_{k=1}^{1}c^{k}_{n}=c^{1}_{n}$ by claim 1 of Properties of Finite Sums. Let $m\in\mathcal{A}$ and suppose $S(m)\in[d]$. Since $m<S(m)$ and $S(m)\le d$, claims 1, 4 and 5 of Properties of the Order on the Natural Numbers give $m\in[d]$, so $\bigl(\sum_{k=1}^{m}c^{k}_{n}\bigr)_{n}$ converges to $0$; by the recursion in claim 1 of Properties of Finite Sums, $\sum_{k=1}^{S(m)}c^{k}_{n}=\sum_{k=1}^{m}c^{k}_{n}+c^{S(m)}_{n}$, which converges to $0+0=0$ by claim 1 of Arithmetic of Limits of Real Sequences. So $S(m)\in\mathcal{A}$, and $\mathcal{A}=\mathbb{N}$ by Principle of Induction for the Natural Numbers. Since $d\in[d]$ (claim 1 of Properties of the Order on the Natural Numbers), the case $m=d$ is the assertion.

Step 12. Property (e). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By (H), the map $\rho\mapsto D^{2}\phi_{\rho}(0_{\mathbb{R}^{d}})$ from $\mathcal{P}_{2}(\mathbb{R}^{d})$ to $\mathcal{S}(d)$ is $\Phi$. We show that $\Phi$ is continuous at $\mu$ relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, for the metric $W_{2}$ and the metric $d_{\mathcal{S}(d)}$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §matrices. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion, read with $X=A=\mathcal{P}_{2}(\mathbb{R}^{d})$ and $Y=\mathcal{S}(d)$, it suffices to let $(\mu_{n})_{n\in\mathbb{N}}$ converge to $\mu$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ and to show that $(\Phi(\mu_{n}))_{n}$ converges to $\Phi(\mu)$ in $(\mathcal{S}(d),d_{\mathcal{S}(d)})$. As in Step 4, $\lim_{n\to\infty}W_{2}(\mu_{n},\mu)=0$. For $i,j\in[d]$ the function $h_{ij}=\partial_{i}\partial_{j}f$ is continuous (Step 1) and $|h_{ij}(x)|\le M\le M(1+\lVert x\rVert^{2})$ for every $x$, by the hypothesis, $1\le1+\lVert x\rVert^{2}$ (Step 1, proof of (G)) and claim 5 of Elementary Arithmetic in an Ordered Field. So Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §quadratic, read with $m=d$ and $h=h_{ij}$, gives that $c^{ij}_{n}=\int h_{ij}\,d\mu_{n}-\int h_{ij}\,d\mu$ satisfies $|c^{ij}_{n}-0|<\varepsilon$ for $n$ large, for every positive $\varepsilon$; as $\bigl||c^{ij}_{n}|-0\bigr|=|c^{ij}_{n}|$ by claim 1 of Properties of the Absolute Value in an Ordered Field, the sequence $(|c^{ij}_{n}|)_{n}$ converges to $0$ (Limit of a Sequence of Real Numbers). By Step 11 applied, for each $i\in[d]$, to the sequences $(|c^{ij}_{n}|)_{n}$ indexed by $j\in[d]$, the sequence $t^{i}_{n}=\sum_{j=1}^{d}|c^{ij}_{n}|$ converges to $0$, and by Step 11 applied to the sequences $(t^{i}_{n})_{n}$ indexed by $i\in[d]$, the sequence $s_{n}=\sum_{i=1}^{d}t^{i}_{n}$ converges to $0$. The matrix $E_{n}=\Phi(\mu_{n})-\Phi(\mu)$ lies in $\mathcal{S}(d)$ by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric and has entries $c^{ij}_{n}$ by Difference of Real Matrices, so Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix §entry-sum gives $\lVert E_{n}\rVert\le s_{n}$. Let $\varepsilon$ be positive and choose $N$ with $|s_{n}-0|<\varepsilon$ for $n\ge N$; for such $n$, by claim 3 of Properties of the Absolute Value in an Ordered Field and claim 2 of Elementary Order Arithmetic in an Ordered Field,

$$d_{\mathcal{S}(d)}\bigl(\Phi(\mu_{n}),\Phi(\mu)\bigr)=\lVert E_{n}\rVert\le s_{n}\le|s_{n}|<\varepsilon .$$

So $(\Phi(\mu_{n}))_{n}$ converges to $\Phi(\mu)$ (Convergent Sequence in a Metric Space), and $\Phi$ is continuous at $\mu$. Now let $\varepsilon$ be positive; by Continuous Map Between Metric Spaces there is a positive $\theta$ such that every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\mu,\nu)<\theta$ satisfies $d_{\mathcal{S}(d)}(\Phi(\nu),\Phi(\mu))<\varepsilon$. Since $W_{2}(\nu,\mu)=W_{2}(\mu,\nu)$ by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry, and $\Phi(\nu)=D^{2}\phi_{\nu}(0_{\mathbb{R}^{d}})$, $\Phi(\mu)=D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})$ by (H), this $\theta$ is as required in property (e).

Step 13. Conclusion. By Steps 4, 5, 6, 10 and 12, $\varphi$ has properties (a) to (e) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test with $Q=\mathcal{P}_{2}(\mathbb{R}^{d})$, so it is an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ its gradient along couplings is $\nabla\varphi(\mu)=Df$ by Step 5, and by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian and (H) its translation Hessian is $H_{\varphi}(\mu)=D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})=\int_{\mathbb{R}^{d}}D^{2}f\,d\mu$. This proves claim 2. Claim 1 was proved in Step 2 (with the bounds (G1) and (G2) of Step 1) and claim 3 in Step 2a. $\blacksquare$