Proof of Sign Reversal for Second-Order Equations on the Wasserstein Space: Subsolutions of F are Supersolutions of its Reversal

Each result cited is universally quantified over the data in its own statement. Throughout, $-x$ denotes the multiple $(-1)x$ in each of $\mathbb{R}$, $L^{2}(\nu;\mathbb{R}^{d})$ and $\mathcal{S}(d)$, so that $-(-x)=x$ and $-(x-y)=(-x)+y$ there, by claim 2 of Zero Products and Elementary Identities in a Field in $\mathbb{R}$, by Elementary Identities in a Vector Space in $L^{2}(\nu;\mathbb{R}^{d})$, and entrywise by Scalar Multiple of a Real Matrix, Sum of Real Matrices and Difference of Real Matrices in $\mathcal{S}(d)$; also $-0=0$ by claims 1 and 2 of Zero Products and Elementary Identities in a Field.

Claim 1. Let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$. Then $-q\in L^{2}(\nu;\mathbb{R}^{d})$, that space being a real vector space by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), so $(\nu,-q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$ by The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts (i); and $-r\in\mathbb{R}$, $-Y\in\mathcal{S}(d)$ as recorded in the statement. Hence $\tilde{F}(\nu,r,q,Y)=-F(\nu,-r,-q,-Y)$ is a real number for all such data, and $\tilde{F}$ is a second-order equation operator over $\mathcal{D}_{\Sigma}$ by The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts (ii). Writing $\tilde{\tilde{F}}$ for the operator obtained from $\tilde{F}$ by the same construction,

$$\tilde{\tilde{F}}(\nu,r,q,Y)=-\tilde{F}(\nu,-r,-q,-Y)=-\bigl(-F(\nu,r,q,Y)\bigr)=F(\nu,r,q,Y).$$

Claim 2. Let $X,Y\in\mathcal{S}(d)$ satisfy $X\preceq Y$ and let $z\in\mathbb{R}^{d}$. By claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum and the bilinearity of the dot product recorded in The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite,

$$z\cdot\bigl((-Y)z\bigr)=z\cdot\bigl(-(Yz)\bigr)=-\bigl(z\cdot(Yz)\bigr),$$

and the same with $X$ in place of $Y$. By Euclidean Space and Real Matrices: Standing Notation (iv) one has $z\cdot(Xz)\le z\cdot(Yz)$, so claim 3 of Elementary Arithmetic in an Ordered Field, used in both directions, gives $-\bigl(z\cdot(Yz)\bigr)\le-\bigl(z\cdot(Xz)\bigr)$; as $z$ was arbitrary, $-Y\preceq-X$. Suppose now that $F$ is degenerate elliptic, and let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$. By Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space (i) applied to $-Y\preceq-X$, $F(\nu,-r,-q,-X)\le F(\nu,-r,-q,-Y)$, and claim 3 of Elementary Arithmetic in an Ordered Field gives

$$\tilde{F}(\nu,r,q,Y)=-F(\nu,-r,-q,-Y)\le-F(\nu,-r,-q,-X)=\tilde{F}(\nu,r,q,X),$$

so $\tilde{F}$ is degenerate elliptic. Conversely, if $\tilde{F}$ is degenerate elliptic, the implication just proved, applied to $\tilde{F}$, shows that $\tilde{\tilde{F}}$ is degenerate elliptic; and $\tilde{\tilde{F}}=F$ by claim 1.

Claim 3. Let $\delta\in\mathbb{R}$ be positive, $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts (iii) and the definition of $\tilde{F}$,

$$\tilde{F}^{+}_{\delta}(\nu,r,q,Y)=\tilde{F}\bigl(\nu,\,r-\delta\,\mathcal{E}(\nu),\,q-\delta\,\Sigma(\nu),\,Y\bigr)=-F\bigl(\nu,\,(-r)+\delta\,\mathcal{E}(\nu),\,(-q)+\delta\,\Sigma(\nu),\,-Y\bigr)=-F^{-}_{\delta}(\nu,-r,-q,-Y),$$

the second equality by the identity $-(x-y)=(-x)+y$ of the preamble and the third by The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts (iii) again. Exchanging the two shifts gives $\tilde{F}^{-}_{\delta}(\nu,r,q,Y)=-F^{+}_{\delta}(\nu,-r,-q,-Y)$ in the same way.

Claim 4. Let $u$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. By Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space (ii) the function $-u$ is a test function with $\nabla(-u)(\nu)=-\nabla u(\nu)$ and $H_{-u}(\nu)=-H_{u}(\nu)$ for every $\nu$. Hence, for $\nu\in\mathcal{D}_{\Sigma}$,

$$\tilde{F}\bigl(\nu,(-u)(\nu),\nabla(-u)(\nu),H_{-u}(\nu)\bigr)=\tilde{F}\bigl(\nu,-u(\nu),-\nabla u(\nu),-H_{u}(\nu)\bigr)=-F\bigl(\nu,u(\nu),\nabla u(\nu),H_{u}(\nu)\bigr),$$

using $-(-x)=x$ three times. By claim 3 of Elementary Arithmetic in an Ordered Field, used in both directions, and $-0=0$, the inequality $F(\nu,u(\nu),\nabla u(\nu),H_{u}(\nu))\le0$ holds if and only if $0\le\tilde{F}(\nu,(-u)(\nu),\nabla(-u)(\nu),H_{-u}(\nu))$. Since this holds for each $\nu\in\mathcal{D}_{\Sigma}$ separately, $u$ is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical supersolution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$, by Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space (i) and Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space (ii). The second equivalence is obtained by reading the same chain in the opposite order, and the third follows from the two by Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space (iii).

Claim 5. We first prove: if $w:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ is a viscosity subsolution of $F$ relative to the penalty pair, then $-w$ is a viscosity supersolution of $\tilde{F}$ relative to the penalty pair.

By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (i) the function $w$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, so by Basic Properties of the Delta-Envelopes on the Wasserstein Space (ii) the function $-w$ is bounded below near each point and $(-w)^{+}_{\delta}=-\,w^{-}_{\delta}$ on $\mathcal{D}$ for every positive $\delta\in\mathbb{R}$. Let $\delta\in\mathbb{R}$ be positive, let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\hat{\mu}\in\mathcal{D}$ be a point at which the function on $\mathcal{D}$ with value $(-w)^{+}_{\delta}(\mu)-\varphi(\mu)$ at $\mu$ has a local minimum relative to $\mathcal{D}$, and let $\varepsilon\in\mathbb{R}$ be positive. By Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space (ii) the function $-\varphi$ is a test function with $\nabla(-\varphi)(\hat{\mu})=-\nabla\varphi(\hat{\mu})$ and $H_{-\varphi}(\hat{\mu})=-H_{\varphi}(\hat{\mu})$. For $\mu\in\mathcal{D}$,

$$(-w)^{+}_{\delta}(\mu)-\varphi(\mu)=-w^{-}_{\delta}(\mu)-\varphi(\mu)=-\bigl(w^{-}_{\delta}(\mu)-(-\varphi)(\mu)\bigr),$$

so by claim 3 of Elementary Arithmetic in an Ordered Field, used in both directions, $\hat{\mu}$ is a point at which the function with value $w^{-}_{\delta}(\mu)-(-\varphi)(\mu)$ at $\mu$ has a local maximum relative to $\mathcal{D}$.

Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (i) to $\delta$, the test function $-\varphi$, the point $\hat{\mu}$ and the number $\varepsilon$ gives $\nu\in\mathcal{D}_{\Sigma}$, $\pi\in\Pi(\nu,\hat{\mu})$, $s\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$ with

$$I(\pi)<\varepsilon^{2},\qquad\bigl|w^{-}_{\delta}(\nu)-w^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,\qquad\bigl|s-w^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,$$ $$\int_{\mathbb{R}^{d+d}}\bigl\lVert q(x)-\nabla(-\varphi)(\hat{\mu})(y)\bigr\rVert^{2}\,\pi(dz)<\varepsilon^{2},\qquad\bigl\lVert Y-H_{-\varphi}(\hat{\mu})\bigr\rVert<\varepsilon,\qquad F^{-}_{\delta}(\nu,s,q,Y)\le\varepsilon ,$$

where $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$. We take $\nu$, $\pi$, $-s$, $-q$ and $-Y$ as the data required of $-w$ at $\hat{\mu}$.

The cost $I(\pi)$ is unchanged. Since $(-w)^{+}_{\delta}=-w^{-}_{\delta}$ on $\mathcal{D}$ and $\nu,\hat{\mu}\in\mathcal{D}$,

$$\bigl|(-w)^{+}_{\delta}(\nu)-(-w)^{+}_{\delta}(\hat{\mu})\bigr|=\bigl|w^{-}_{\delta}(\nu)-w^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,\qquad\bigl|-s-(-w)^{+}_{\delta}(\hat{\mu})\bigr|=\bigl|s-w^{-}_{\delta}(\hat{\mu})\bigr|<\varepsilon,$$

by claim 2 of Properties of the Absolute Value in an Ordered Field. A representative of $-q$ takes the value $-q(x)$ at $x$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (ii), so for every $z$,

$$\bigl\lVert -q(x)-\nabla\varphi(\hat{\mu})(y)\bigr\rVert^{2}=\bigl\lVert -\bigl(q(x)-\nabla(-\varphi)(\hat{\mu})(y)\bigr)\bigr\rVert^{2}=\bigl\lVert q(x)-\nabla(-\varphi)(\hat{\mu})(y)\bigr\rVert^{2},$$

the second equality by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and claim 2 of Properties of the Absolute Value in an Ordered Field; hence the discrepancy of $-q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ equals that of $q$ and $\nabla(-\varphi)(\hat{\mu})$ along $\pi$, and is smaller than $\varepsilon^{2}$. Likewise $-Y-H_{\varphi}(\hat{\mu})=-\bigl(Y-H_{-\varphi}(\hat{\mu})\bigr)$, so $\lVert -Y-H_{\varphi}(\hat{\mu})\rVert=\lVert Y-H_{-\varphi}(\hat{\mu})\rVert<\varepsilon$ by the identity $\lVert -B\rVert=\lVert B\rVert$ recorded in the statement. Finally, claim 3 applied at the data $(-s,-q,-Y)$ gives $\tilde{F}^{+}_{\delta}(\nu,-s,-q,-Y)=-F^{-}_{\delta}(\nu,s,q,Y)$, and $F^{-}_{\delta}(\nu,s,q,Y)\le\varepsilon$ gives $-\varepsilon\le-F^{-}_{\delta}(\nu,s,q,Y)$ by claim 3 of Elementary Arithmetic in an Ordered Field. All the requirements of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (ii) are met, so $-w$ is a viscosity supersolution of $\tilde{F}$ relative to the penalty pair.

The companion implication — if $w$ is a viscosity supersolution of $F$ relative to the penalty pair, then $-w$ is a viscosity subsolution of $\tilde{F}$ relative to the penalty pair — is proved by the same argument with the following exchanges: Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (ii) in place of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (i) and conversely; the second clause of Basic Properties of the Delta-Envelopes on the Wasserstein Space (ii) in place of the first, giving $(-w)^{-}_{\delta}=-\,w^{+}_{\delta}$; local maxima in place of local minima and conversely; and the second identity of claim 3 in place of the first, with the final inequality $-\varepsilon\le F^{+}_{\delta}(\nu,s,q,Y)$ giving $\tilde{F}^{-}_{\delta}(\nu,-s,-q,-Y)=-F^{+}_{\delta}(\nu,s,q,Y)\le\varepsilon$.

Now let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. If $u$ is a viscosity subsolution of $F$, the first implication gives that $-u$ is a viscosity supersolution of $\tilde{F}$. If conversely $-u$ is a viscosity supersolution of $\tilde{F}$, the companion implication applied to the function $-u$ and the operator $\tilde{F}$ gives that $-(-u)=u$ is a viscosity subsolution of $\tilde{\tilde{F}}$, which is $F$ by claim 1. This is the first equivalence, and the second is obtained by exchanging the two implications in the same way. The third follows from the two by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space (iii). In each equivalence the local boundedness required of $u$ corresponds to that required of $-u$ by Basic Properties of the Delta-Envelopes on the Wasserstein Space (ii).