Proof of First- and Second-Order Conditions at a Penalised Extremum of a Test Function on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement.

Claim 1. Let $\hat{\mu}\in\mathcal{D}_{\Sigma}$ be a point at which $\chi-\delta\mathcal{E}$ has a local maximum relative to $\mathcal{D}$. By Local Maximum of a Function Relative to a Subset of a Metric Space there is a positive $R\in\mathbb{R}$ such that

The second-order condition. Let $g:\mathbb{R}^{d}\to\mathbb{R}$ be the function $g(a)=\chi\bigl((\tau_{a})_{\#}\hat{\mu}\bigr)$; by First Variations of a Test Function on the Wasserstein Space Along Gradient Displacements and Along Translations §hessian it is of class $C^{2}$ on $\mathbb{R}^{d}$ with Hessian matrix $H_{\chi}(\hat{\mu})$ at $0_{\mathbb{R}^{d}}$. Let $a\in\mathbb{R}^{d}$ satisfy $\lVert a\rVert<R$. By Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation-invariant the measure $(\tau_{a})_{\#}\hat{\mu}$ belongs to $\mathcal{D}$ and $\mathcal{E}\bigl((\tau_{a})_{\#}\hat{\mu}\bigr)=\mathcal{E}(\hat{\mu})$, and by First Variations of a Test Function on the Wasserstein Space Along Gradient Displacements and Along Translations §translation $W_{2}\bigl((\tau_{a})_{\#}\hat{\mu},\hat{\mu}\bigr)\le\lVert a\rVert<R$. So $(\ast)$ applies to $\mu=(\tau_{a})_{\#}\hat{\mu}$ and reads $g(a)-\delta\,\mathcal{E}(\hat{\mu})\le g(0_{\mathbb{R}^{d}})-\delta\,\mathcal{E}(\hat{\mu})$, whence $g(a)\le g(0_{\mathbb{R}^{d}})$ by the compatibility of the order with addition, an axiom of Ordered Field. Since $d_{E}(a,0_{\mathbb{R}^{d}})=\lVert a\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and $\mathbb{R}^{d}$ is open in itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, the function $g$ has a local maximum at $0_{\mathbb{R}^{d}}$ relative to $\mathbb{R}^{d}$, so claim 1 of First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$ gives $H_{\chi}(\hat{\mu})\preceq0_{d}$.

The first-order condition. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. By Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §variation there is a positive $t_{0}\in\mathbb{R}$ such that $(\mathrm{id}+t\,\nabla\psi)_{\#}\hat{\mu}\in\mathcal{D}$ for every $t\in(-t_{0},t_{0})$ and such that the function $e:(-t_{0},t_{0})\to\mathbb{R}$, $e(t)=\mathcal{E}\bigl((\mathrm{id}+t\,\nabla\psi)_{\#}\hat{\mu}\bigr)$, is differentiable at $0$ with $e'(0)=\langle\Sigma(\hat{\mu}),\nabla\psi\rangle_{\hat{\mu}}$. By First Variations of a Test Function on the Wasserstein Space Along Gradient Displacements and Along Translations §gradient the function $k:(-t_{0},t_{0})\to\mathbb{R}$, $k(t)=\chi\bigl((\mathrm{id}+t\,\nabla\psi)_{\#}\hat{\mu}\bigr)$, is differentiable at $0$ with $k'(0)=\langle\nabla\chi(\hat{\mu}),\nabla\psi\rangle_{\hat{\mu}}$. The point $0$ is interior to $(-t_{0},t_{0})$ by Basic Facts about Intervals of the Real Line and Their Interior Points §open-interval, since $-t_{0}<-0=0<t_{0}$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, so by claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives the function $h=k+(-\delta)e$, whose value at $t$ is $k(t)-\delta\,e(t)$ by claim 2 of Zero Products and Elementary Identities in a Field, is differentiable at $0$ with

the second equality by Elementary Identities in a Real Inner Product Space §bilinear.

Put $c=1+\lVert\nabla\psi\rVert_{\hat{\mu}}$, positive by claim 3 of Elementary Order Arithmetic in an Ordered Field applied to $0<1$ (claim 6 of that lemma) and $0\le\lVert\nabla\psi\rVert_{\hat{\mu}}$, and let $\rho$ be the lesser of $t_{0}$ and $R\,c^{-1}$ (claim 9 of that lemma), positive because it is one of them, $c^{-1}$ being positive by claim 7 and $R\,c^{-1}$ by claim 5. For $t\in(-t_{0},t_{0})$ with $|t|<\rho$, The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §distance and claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $|t|$, followed by claim 10 of Elementary Order Arithmetic in an Ordered Field with the positive multiplier $c$, give

Hence $(\ast)$ applies to $\mu=(\mathrm{id}+t\,\nabla\psi)_{\#}\hat{\mu}$, which lies in $\mathcal{D}$, and yields $h(t)\le h(0)$ for every such $t$; that is, $h$ has a local maximum at $0$ relative to $(-t_{0},t_{0})$. By Vanishing of the Derivative at an Interior Local Extremum, $h'(0)=0$, so

Write $v=\nabla\chi(\hat{\mu})-\delta\,\Sigma(\hat{\mu})$. The field $\nabla\chi(\hat{\mu})$ lies in $T_{\hat{\mu}}$ by Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §gradient and $\Sigma(\hat{\mu})$ lies in $T_{\hat{\mu}}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so $v\in T_{\hat{\mu}}$, that set being a linear subspace of $L^{2}(\hat{\mu};\mathbb{R}^{d})$ 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. The function $\ell:C_{c}^{\infty}(\mathbb{R}^{d})\to\mathbb{R}$ with constant value $0$ satisfies the hypotheses of claim 4 of Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients with $C=0$, since $|\ell(\psi)|=0=0\cdot\lVert\nabla\psi\rVert_{\hat{\mu}}$ by claim 1 of Properties of the Absolute Value in an Ordered Field and claim 1 of Zero Products and Elementary Identities in a Field. Both $v$ and the zero element of $T_{\hat{\mu}}$ satisfy $\langle\cdot,\nabla\psi\rangle_{\hat{\mu}}=\ell(\psi)$ for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, the latter by Elementary Identities in a Real Inner Product Space §zero; by the uniqueness in that claim they are equal, so $v$ is the zero element; adding $\delta\,\Sigma(\hat{\mu})$ to both sides of $\nabla\chi(\hat{\mu})+\bigl(-\delta\,\Sigma(\hat{\mu})\bigr)=0$, which is $v=0$ written out with claim 2 of Elementary Identities in a Vector Space, gives $\nabla\chi(\hat{\mu})=\delta\,\Sigma(\hat{\mu})$ by the associativity, inverse and identity axioms of the vector space $L^{2}(\hat{\mu};\mathbb{R}^{d})$.

Claim 2. Let $\hat{\mu}\in\mathcal{D}_{\Sigma}$ be a point at which $\chi+\delta\mathcal{E}$ has a local minimum relative to $\mathcal{D}$. By Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space §difference the function $-\chi$ is a test function with $\nabla(-\chi)(\nu)=-\nabla\chi(\nu)$ and $H_{-\chi}(\nu)=-H_{\chi}(\nu)$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\mu\in\mathcal{D}$ one has $-\bigl(\chi(\mu)+\delta\,\mathcal{E}(\mu)\bigr)=(-\chi)(\mu)-\delta\,\mathcal{E}(\mu)$ by the distributivity axiom of the field $\mathbb{R}$ and claim 2 of Zero Products and Elementary Identities in a Field, so the function $(-\chi)-\delta\mathcal{E}$ on $\mathcal{D}$ is the negative of $\chi+\delta\mathcal{E}$; by claim 3 of Elementary Arithmetic in an Ordered Field, used in both directions, the inequalities defining a local minimum of $\chi+\delta\mathcal{E}$ at $\hat{\mu}$ relative to $\mathcal{D}$ are equivalent to those defining a local maximum of $(-\chi)-\delta\mathcal{E}$ there. Claim 1, applied to the test function $-\chi$, therefore gives

Multiplying the first identity by $-1$ and using $(-1)\bigl((-1)w\bigr)=w$, which follows from claim 5 of Elementary Identities in a Vector Space and the scalar-multiplication axioms of the vector space $L^{2}(\hat{\mu};\mathbb{R}^{d})$, gives $\nabla\chi(\hat{\mu})=-\delta\,\Sigma(\hat{\mu})$. For the second, $-H_{\chi}(\hat{\mu})=0_{d}-H_{\chi}(\hat{\mu})$ entrywise by Difference of Real Matrices, and $0_{d}-H_{\chi}(\hat{\mu})\preceq0_{d}$ holds if and only if $0_{d}\preceq H_{\chi}(\hat{\mu})$ by Comparison with the Zero Matrix in the Positive Semidefinite Ordering.