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

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, defined on $\mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$. In this statement the letter $r$ denotes a real number; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used. For $\nu\in\mathcal{D}_{\Sigma}$ and $q\in L^{2}(\nu;\mathbb{R}^{d})$, $-q$ denotes the real multiple $(-1)q$; for $Y\in\mathcal{S}(d)$, $-Y$ denotes the real multiple $(-1)Y$ of Euclidean Space and Real Matrices: Standing Notation (ii), which belongs to $\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure and satisfies $\lVert -Y\rVert=\lVert Y\rVert$ by claim 5 of Properties of the Norm of a Symmetric Real Matrix applied with the multiplier $-1$, whose absolute value is $1$ by claim 2 of Properties of the Absolute Value in an Ordered Field and Absolute Value in an Ordered Field, since $0\le1$ by claim 1 of Elementary Arithmetic in an Ordered Field; and for $u:\mathcal{D}\to\mathbb{R}$, $-u$ denotes the function on $\mathcal{D}$ with value $-u(\nu)$ at $\nu$, and likewise for a function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. Intrinsic test functions, degenerate ellipticity, the classical subsolutions, supersolutions and solutions of an operator on a set of measures, and the viscosity subsolutions, supersolutions and solutions relative to a penalty pair are those of the definitions cited. Define

$$\tilde{F}:\ \mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)\to\mathbb{R},\qquad \tilde{F}(\nu,r,q,Y)=-F(\nu,-r,-q,-Y).$$

Then the following hold.

1. (Operator)(i) $\tilde{F}$ is a second-order equation operator over $\mathcal{D}_{\Sigma}$, and applying the same construction to $\tilde{F}$ returns $F$.

2. (Ellipticity)(ii) $\tilde{F}$ is degenerate elliptic if and only if $F$ is degenerate elliptic.

3. (Shifts)(iii) For every positive $\delta\in\mathbb{R}$, every $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, every $r\in\mathbb{R}$ and every $Y\in\mathcal{S}(d)$,

$$\tilde{F}^{+}_{\delta}(\nu,r,q,Y)=-F^{-}_{\delta}(\nu,-r,-q,-Y),\qquad \tilde{F}^{-}_{\delta}(\nu,r,q,Y)=-F^{+}_{\delta}(\nu,-r,-q,-Y).$$

4. (Classical solutions)(iv) Let $u$ be an intrinsic test function on $\mathcal{D}_{\Sigma}$. Then $u$ is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical supersolution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$; $u$ is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical subsolution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$; and $u$ is a classical solution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical solution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$.

5. (Viscosity solutions)(v) Let $u:\mathcal{D}\to\mathbb{R}$. Then $u$ is a viscosity subsolution of $F$ relative to the penalty pair if and only if $-u$ is a viscosity supersolution of $\tilde{F}$ relative to the penalty pair; $u$ is a viscosity supersolution of $F$ relative to the penalty pair if and only if $-u$ is a viscosity subsolution of $\tilde{F}$ relative to the penalty pair; and $u$ is a viscosity solution of $F$ relative to the penalty pair if and only if $-u$ is a viscosity solution of $\tilde{F}$ relative to the penalty pair. In each equivalence the penalty-subordinate growth required of $u$ by the one definition holds if and only if that required of $-u$ by the other holds.